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Bias-Aware Inference in Regularized Regression Models
\maketitle
\begin{abstract}
We consider inference on a scalar regression coefficient under a constraint on
the magnitude of the control coefficients. A class of estimators based on a
regularized propensity score regression is shown to exactly solve a tradeoff
between worst-case bias and variance. We derive \acp{CI} based on these
estimators that are bias-aware: they account for the possible bias of the
estimator. Under homoskedastic Gaussian errors, these estimators and \acp{CI}
are near-optimal in finite samples for \acl{MSE} and \ac{CI} length. We also
provide conditions for asymptotic validity of the \acp{CI} with unknown and
possibly heteroskedastic error distribution, and derive novel optimal rates of
convergence under high-dimensional asymptotics that allow the number of
regressors to increase more quickly than the number of observations. Extensive
simulations and an empirical application illustrate the performance of our
methods.
\end{abstract}
\clearpage
\section{Introduction}
We are interested in estimation and inference on a scalar coefficient $\beta$ in
a linear regression model
\begin{equation}\label{linear_regression_eq}
Y_{i} = w_{i}\beta + z_{i}'\gamma + \varepsilon_{i}, \qquad i=1, \dotsc, n,
\end{equation}
when the $k$-vector $z_i$ of controls is large. In such settings, the classic
\ac{OLS} estimator is often uninformative, exhibiting variance that is too
large; the estimator is not even defined when $k>n$. This motivates modifying
the \ac{OLS} objective function to penalize large values of $\gamma$, thereby
lowering variance at the cost of introducing bias.
The most popular of these approaches is the lasso
\citep{tibshirani_regression_1996} or other variants of $\ell_{1}$ penalization
\citep*[e.g.][]{CaTa07,BeChWa11}. There is a large literature \citep[see,
e.g.][for a review]{buhlmann_statistics_2011} showing favorable \ac{MSE}
properties of these estimators when $\gamma$ is sparse. For inference, several
papers have proposed \acp{CI} based on ``double lasso'' estimators \citep[see,
among
others,][]{belloni_inference_2014,javanmard_confidence_2014,van_de_geer_asymptotically_2014,zhang_confidence_2014},
with asymptotic justification relying on rate conditions for the sparsity of
$\gamma$. However, in many applications in economics, the sparsity assumption is
not compelling and may be hard to motivate. Furthermore, it is unclear what
sparsity level this approach implicitly imposes in a given finite sample.
We propose bounding the \emph{magnitude} of the control coefficients, rather
than their sparsity level, by assuming that $\operatorname{Pen}(\gamma)\leq C$. The penalty
function $\operatorname{Pen}(\cdot)$ formalizes the notion of magnitude, and it can
incorporate any restrictions on $\gamma$ that place it in a convex symmetric
set. Such restrictions arise naturally in a plethora of applications. For
instance, the dimensionality of the control vector is often large due to the
inclusion of additional controls that are collectively believed to only be
weakly associated with the outcome, but are nonetheless included to purge any
possible confounding. One can then take the penalty to be an $\ell_{p}$ norm for
the additional controls. If $z_i'\gamma$ is a basis approximation to some smooth
function, we can define $\operatorname{Pen}(\gamma)$ to incorporate bounds on the derivatives
of this function. The regularity parameter $C$ plays a role analogous to a
sparsity bound.
We obtain sharp finite-sample results deriving near-optimal estimators and
\acp{CI} under this penalty constraint and the idealized assumption that the
regression errors $\varepsilon_{i}$ are Gaussian with a known homoskedastic
variance. We show that the class of estimators that exactly resolves the
trade-off between worst-case bias and variance can be obtained by (1) running a
penalized propensity score regression of $w_i$ on $z_i$ using $\operatorname{Pen}(\cdot)$ as
the penalty function; and then (2) using the residuals from this regression as
an instrument in the univariate regression of $Y_{i}$ on $w_{i}$. \acp{CI} based
on these estimators can be constructed by using a critical value that
incorporates the worst-case bias of the estimator, which we show obtains
automatically as a byproduct of the regularized regression in step (1). Because
these \acp{CI} are bias-aware---they account for the potential finite-sample
bias of the estimator---they are valid in finite samples in this idealized
Gaussian setup. We show how to choose the weight on the penalty function in step
(1) to optimize the \ac{MSE} of the resulting estimator, or the length of the
resulting \ac{CI}.
In the more realistic setting with heteroskedasticity and an unknown error
distribution, bias-aware \acp{CI} can be formed using heteroskedasticity-robust
variance estimators, and we give conditions for their asymptotic validity,
allowing for high-dimensional asymptotics with $k\gg n$. Our setup can allow for
the effect of $w_{i}$ on the outcome to be heterogeneous, either by including
interactions of the treatment and demeaned covariates among the controls, or by
reinterpreting $\beta$ in \cref{linear_regression_eq} as a weighted average
treatment effect. We show that the treatment weights solve a bias-variance
tradeoff in a problem where we can pick the estimand to make the estimation
problem as easy as possible.
We also employ the high-dimensional asymptotics to study rates of convergence of the
bias-aware \acp{CI} when $\operatorname{Pen}(\gamma)$ is an $\ell_p$ norm. We show that if
$k\gg n$ and $C$ does not shrink with $n$, the optimal \ac{CI} shrinks more
slowly than $n^{-1/2}$, so that the bias term asymptotically dominates;
accounting for bias in \ac{CI} construction thus cannot be avoided even in large
samples. Furthermore, we show that, in the $\ell_1$ case, this rate cannot be
improved even if one additionally imposes the same $\ell_1$ bound in the
propensity score regression of $w_i$ on $z_i$, as well as a certain degree of
sparsity in both regressions.
Explicit specification of the regularity parameter $C$ that bounds the magnitude
of $\gamma$ is a key input for our approach. Our efficiency bounds show that it
is impossible to automate the choice of $C$ when forming \acp{CI}. We discuss
how relating the magnitude of $\gamma$ to other quantities, such as the
magnitude of the control coefficients in a short regression that only includes
baseline controls, can help guide its choice. We develop a rule of thumb
specification for $C$ based on this idea that we use in our simulations and
empirical application. Robustness of the results can be assessed by computing a
breakdown value of $C$, its largest value such that the empirical finding of
interest, such as rejecting a particular null hypothesis, holds. Selection of
$C$ cannot be automated due to the impossibility of getting a sufficiently
informative data-driven upper bound for it. We show, however, that it is possible
to obtain a \emph{lower} \ac{CI} for $C$, which can be used as a specification
check to ensure that the chosen value is not too low.
The requirement to explicitly choose $C$ may seem like a limitation of our
approach relative to sparsity-based approaches, where the analogous tuning
parameter, the degree of sparsity, does not need to be explicitly specified.
However, good finite-sample performance of such methods relies on bounding these
tuning parameters implicitly, and such implicit bounds are hard to calculate or
evaluate in a given problem. We demonstrate this issue in a Monte Carlo
analysis, where we show that double-lasso \acp{CI} suffer from moderate to
severe undercoverage even in designs that are apparently sparse. Indeed, we view
the explicit specification of $C$ as an advantage of our approach, because our
coverage guarantees and efficiency bounds are based on transparent assumptions
rather than ``asymptotic promises'' about tuning parameters that are hard to
evaluate in a particular sample.
Our results relate to several strands of literature. Our procedures and
efficiency bounds apply the general theory of estimation and inference on linear
functionals in convex Gaussian models developed in \citet{IbKh85},
\citet{donoho94}, \citet{low95} and \citet{ArKo18optimal}, and add to a growing
literature applying this approach to various settings, including
\citet{ArKo20ate,ArKo20sensitivity}, \citet{kolesar_inference_2018},
\citet{ImWa19}, \citet{RaRo19}, \citet{noack_bias-aware_2019}, and
\citet{kwon_inference_2020}. \citet{muralidharan_factorial_2020} apply the
approach in the present paper to experiments with factorial designs and bounds
on interaction effects.
The idea of using propensity score residuals to estimate $\beta$ goes back at
least to the work of \citet{robinson_root-n-consistent_1988} on the partly
linear model. We provide a novel finite-sample justification for this idea, as
well as an exact result giving the optimal penalization of this regression. Our
setup allows for a general form of $\operatorname{Pen}(\cdot)$, and yields existing estimators
in a few special cases; the bias-aware \acp{CI} to accompany such estimators are
novel. First, we recover the optimal linear estimators in
\citet{heckman_minimax_1988}, who considered the partly linear model with a
penalty function bounding the first or second derivative of a univariate
nonparametric regression function. Next, we reproduce the result in \citet{li82}
that the optimal estimator uses ridge regression when the penalty corresponds to
an $\ell_{2}$ norm. Finally, \citet{li_linear_2020} consider the weighted
$\ell_2$ norm $\operatorname{Pen}(\gamma)=\left(\sum_{i=1}^n (z_i'\gamma)^{2} \right)^{1/2}$.
They develop bias-aware \acp{CI} under this penalty based on a likelihood ratio
statistic, which are numerically shown to be close to optimal under
homoskedasticity and a particular weighted average length criterion. However,
unlike our \ac{CI}, the \citeauthor{li_linear_2020} \ac{CI} may end up being longer
than the long regression \ac{CI}, as we illustrate in our empirical application
in \Cref{sec:empirical-app}.
The next section presents our finite-sample results in the idealized model with
Gaussian errors. \Cref{sec:impl-with-non} discusses implementation in the more
realistic setting with unknown error distribution. \Cref{sec:asymptotics}
derives rates of convergence under high-dimensional asymptotics and bounds on an
$\ell_p$ norm. \Cref{sec:nonconvex_efficiency} compares our approach to \acp{CI}
motivated by sparsity constraints. The performance of our methods is evaluated
in a Monte Carlo study in \Cref{sec:simulation-results}, while
\Cref{sec:empirical-app} illustrates them in an empirical application. Proofs
and auxiliary results appear in appendices.
\section{Finite-sample results}\label{sec:finite-sample-result}
This section sets up an idealized version of our model with Gaussian
homoskedastic errors. We then show how to construct estimators and \acp{CI} in
this model that are near-optimal in finite samples.
\subsection{Setup}\label{sec:setup}
We write the model in \cref{linear_regression_eq} in vector form as
\begin{equation}\label{eq:linear_regression_vector}
Y = w\beta + Z\gamma + \varepsilon,
\end{equation}
where $w=(w_1,\dotsc, w_n)'\in\mathbb{R}^n$ is the variable of interest with
coefficient $\beta\in\mathbb{R}$ and
$Z=(z_{1}, \dotsc, z_{n})'\in \mathbb{R}^{n\times k}$ is a matrix of control
variables. The design matrix $X=(w, Z)$ is treated as fixed. To obtain
finite-sample results, we further assume that the errors are normal and
homoskedastic $\varepsilon\sim\mathcal{N}(0, \sigma^{2}I_{n})$, with $\sigma^{2}$ known.
To ensure informative inference on $\beta$ when $k$ is large relative to $n$
(including the case $k>n$), the researcher needs to make \emph{a priori}
restrictions on the control coefficients $\gamma$. We assume that these
restrictions can be formalized by restricting the parameter space for
$(\beta, \gamma')'$ to be $\mathbb{R}\times \Gamma$ where, for some linear
subspace $\mathcal{G}$ of $\mathbb{R}^{k}$ and some seminorm $\operatorname{Pen}(\cdot)$ on $\mathcal{G}$,
\begin{equation}\label{eq:Gamma_def}
\Gamma=\Gamma(\operatorname{Pen};C)= \{\gamma\in \mathcal{G}\colon \operatorname{Pen}(\gamma)\le C\}.
\end{equation}
The requirement that $\operatorname{Pen}(\cdot)$ be a seminorm means that it satisfies the
triangle inequality
($\operatorname{Pen}(\gamma+\tilde\gamma)\le \operatorname{Pen}(\gamma)+\operatorname{Pen}(\tilde\gamma)$), and
homogeneity ($\operatorname{Pen}(c\gamma)=\abs{c}\operatorname{Pen}(\gamma)$ for any scalar $c$), but,
unlike a norm, it is not necessarily positive definite ($\operatorname{Pen}(\gamma)=0$ does not
imply $\gamma=0$). This allows us to cover settings where only a subset of the
control coefficients is restricted.
A common class of restrictions arises when $\operatorname{Pen}(\gamma)$ is a weighted
$\ell_{p}$ norm on a subset of the coefficients. To describe two examples in
this class of restrictions, partition the controls into a set of $k_{1}\geq 0$
unrestricted baseline controls and a set of $k_{2}=k-k_{1}$ additional controls,
$Z=(Z_{1}, Z_{2})$. Partition $\gamma=(\gamma_{1}', \gamma_{2}')'$ accordingly.
Let $H_{A}$ denote the projection matrix onto the column space of a matrix $A$.
Let $\|\cdot\|_p$ denote the $\ell_p$ norm.
\begin{example}[name={$\ell_{2}$ penalty},label=example:l2]
We specify the penalty as
\begin{equation}\label{eq:l2-penalty}
\operatorname{Pen}(\gamma)=\norm{M\gamma}_{2}=\sqrt{\gamma'M'M\gamma},
\end{equation}
where the $k_{2}\times k$ matrix $M$ incorporates scaling the variables and
picking out which variables are to be constrained. If $M=(0,I_{k_{2}})$, then
$\operatorname{Pen}(\gamma)=\norm{\gamma_{2}}_{2}$, with $\gamma_{1}$ unconstrained. Setting
$M=(0,(Z_{2}'(I-H_{Z_{1}})Z_{2}/n)^{1/2})$ corresponds to the specification
considered in \citet{li_linear_2020}, which restricts the average of the
squared mean effects $z_{2i}'\gamma_{2}$ on $Y_{i}$, after controlling for the
baseline controls $z_{1i}$.
\end{example}
\begin{example}[name={$\ell_{1}$ penalty},label=example:l1]
A weighted $\ell_{1}$ penalty replaces the norm in \cref{eq:l2-penalty} with
an $\ell_{1}$ norm. We focus on the unweighted case for simplicity, setting
$\operatorname{Pen}(\gamma)=\norm{\gamma_{2}}_{1}$.
\end{example}
In addition to selecting the penalty, the specification of $\Gamma$ also
requires the researcher to pick the regularity parameter $C$; here we take it as
given, and defer a discussion of its choice to \Cref{sec:impl-with-non}.
Formulating the parameter space $\Gamma$ in terms of a seminorm is not
restrictive in the sense that essentially any convex set $\Gamma$ that is
symmetric ($\gamma\in\Gamma$ implies $-\gamma\in\Gamma$) can be defined in this
way \citep[see][Proposition 5, p.~26]{yosida_functional_1995}. Although we rule
out non-convex constraints on $\Gamma$, such as sparsity, our results
nonetheless have implications for such settings, as we discuss in
\Cref{sec:nonconvex_efficiency}.
Our goal is to construct estimators and \acp{CI} for $\beta$. To evaluate
estimators $\hat{\beta}$ of $\beta$, we consider their worst-case performance
over the parameter space $\mathbb{R}\times \Gamma$ under the \ac{MSE} criterion,
\begin{equation*}
\ensuremath\operatorname{R}_{\textnormal{MSE}}(\hat\beta;\Gamma)
=\sup_{\beta\in\mathbb{R}, \gamma\in\Gamma}E_{\beta, \gamma}[(\hat\beta-\beta)^{2}],
\end{equation*}
where $E_{\beta, \gamma}$ denotes expectation under $(\beta, \gamma')'$. An
interval $\{\hat\beta\pm \hat\chi\}$ with half-length $\hat\chi=\hat\chi(Y, X)$
is a \ac{CI} with level $1-\alpha$ if it satisfies the coverage requirement
\begin{equation*}
\inf_{\beta\in\mathbb{R}, \gamma\in\Gamma}
P_{\beta, \gamma}\left(\beta\in \{\hat\beta\pm \hat\chi\} \right)\ge 1-\alpha,
\end{equation*}
where $P_{\beta, \gamma}$ denotes probability under $(\beta, \gamma')'$. To
compare two CIs under a particular parameter vector $(\beta, \gamma')'$, we
prefer the one with shorter expected length $E_{\beta, \gamma}[2\hat\chi]$. Note
that optimizing expected length will not necessarily lead to \acp{CI} centered
at an estimator $\hat\beta$ that is optimal under the \ac{MSE} criterion.
\subsection{Linear estimators and CIs}\label{sec:line-estim-cis}
We start by considering estimators that are linear in the outcomes $Y$,
$\hat{\beta}=a'Y$, and derive \acp{CI} based on such estimators. The $n$-vector
of weights $a$ may depend on the design matrix $X$ or the known variance
$\sigma^{2}$. In~\Cref{sec:optimal-weights} below, we show how to choose the
weights $a$ optimally, and in \Cref{sec:linear_efficiency} we show that when $a$
is optimally chosen, the resulting estimators and \acp{CI} are optimal or
near-optimal among all procedures, not just linear ones.
Under a given parameter vector $(\beta, \gamma')'$, the bias of
$\hat{\beta}=a'Y$ is given by $a'(w\beta+Z\gamma) - \beta$. As
$(\beta, \gamma')'$ ranges over the parameter space $\mathbb{R}\times \Gamma$,
the bias ranges over the interval
$[-\operatorname{\overline{bias}}_{\Gamma}(\hat\beta), \operatorname{\overline{bias}}_{\Gamma}(\hat\beta)]$, where
\begin{equation}\label{eq:maxbias_def}
\operatorname{\overline{bias}}_{\Gamma}(\hat\beta)
= \sup_{\beta\in\mathbb{R}, \gamma\in \Gamma} a'(w\beta+Z\gamma) - \beta
\end{equation}
denotes the worst-case bias. The variance of $\hat\beta$ does not depend on
$(\beta, \gamma')'$, and is given by $\operatorname{var}(\hat\beta) = \sigma^2 a'a$.
To form a CI centered at $\hat\beta$, note that the $z$-statistic
$(\hat\beta-\beta)/\operatorname{var}(\hat\beta)^{1/2}$ follows a $\mathcal{N}(b,1)$ distribution with
mean bounded by $\abs{b}\le \operatorname{\overline{bias}}_{\Gamma}(\hat\beta)/\operatorname{var}(\hat\beta)^{1/2}$.
Thus, a two-sided \ac{CI} can be formed as
\begin{equation}\label{eq:linear-CI}
\hat{\beta} \pm \chi, \quad
\text{where}\quad
\chi = \operatorname{var}(\hat\beta)^{1/2}
\cdot \operatorname{cv}_{\alpha}\left(\operatorname{\overline{bias}}_{\Gamma}(\hat\beta)/\operatorname{var}(\hat\beta)^{1/2}\right),
\end{equation}
and $\operatorname{cv}_{\alpha}(B)$ denotes the $1-\alpha$ quantile of the folded normal
distribution, $\abs{\mathcal{N}(B,1)}$.\footnote{\label{fn:compute_cv}The critical value
$\operatorname{cv}_{1-\alpha}(B)$ can be computed as the square root of the $1-\alpha$
quantile of a non-central $\chi^{2}$ distribution with $1$ degree of freedom
and non-centrality parameter $B^{2}$.}
This \ac{CI} is \emph{bias-aware} in that the critical value
$\operatorname{cv}_{\alpha}(\cdot)$ reflects the potential finite-sample bias of
$\hat{\beta}$. Following the terminology in \citet{donoho94}, we refer to the
\ac{CI} as a \acf{FLCI}, since its length $2\chi$ is fixed: it depends only on
the non-random design matrix $X$, and known variance $\sigma^{2}$, but not on
$Y$ or the parameter vector $(\beta, \gamma')'$.
\subsection{Optimal weights}\label{sec:optimal-weights}
Both the \ac{MSE}
$R(\hat\beta;\Gamma)= \operatorname{\overline{bias}}_{\Gamma}(\hat\beta)^2 + \operatorname{var}(\hat\beta)$ and the
\ac{FLCI} length $2\chi$ given in~\cref{eq:linear-CI} are increasing in the
variance of $\hat{\beta}$ and in its worst-case bias
$\operatorname{\overline{bias}}_{\Gamma}(\hat{\beta})$. Therefore, to find the optimal weights, we
first minimize variance subject to a bound $B$ on worst-case bias,
\begin{equation}\label{eq:a_optimization}
\min_{a\in\mathbb{R}} a'a
\quad\text{s.t.}\quad
\sup_{\beta\in\mathbb{R}, \gamma\in \Gamma} a'(w\beta+Z\gamma) - \beta
\le B.
\end{equation}
We then vary the bound $B$ to find the optimal bias-variance tradeoff for a
given criterion (\ac{MSE} or \ac{FLCI} length). Since this optimization does not
depend on the outcome data $Y$, optimizing the weights does not affect the
coverage properties of the resulting \ac{CI}.
Our main computational result, in \Cref{thm:optimization_equivalence} below,
shows that the estimator solving the optimization problem in
\cref{eq:a_optimization} is given by a simple two-step procedure. In the first
step, we estimate a penalized regression of $w$ on $Z$ with penalty $\operatorname{Pen}(\pi)$,
so that the coefficient estimate on $Z$, $\pi_\lambda$, solves
\begin{equation}\label{eq:pi_optimization}
\min_{\pi} \norm{w-Z\pi}^{2}_{2} \quad\text{s.t.}\quad \operatorname{Pen}(\pi)\le t_\lambda,
\end{equation}
where $t_\lambda$ is a bound on the penalty term. We refer to
\cref{eq:pi_optimization} as a (regularized) propensity score regression, even
though we don't require $w_{i}$ to be binary. In the second step, we use the
residuals $\tilde{w}_{\lambda}:=w-Z\pi_{\lambda}$ from the propensity score
regression as instruments in a univariate regression of $Y$ on $w$. The tuning
parameter $\lambda$ indexes the weight placed on the constraint in
\cref{eq:pi_optimization}, and its selection depends on the criterion we are
optimizing. It may correspond to the Lagrange multiplier in a Lagrangian
formulation of \cref{eq:pi_optimization}, or, if we can
solve \cref{eq:pi_optimization} directly, we may take $t_\lambda=\lambda$.
\begin{theorem}\label{thm:optimization_equivalence}
Let $\tilde{w}_{\lambda}=w-Z\pi_{\lambda}$, where $\pi_\lambda$ solves
\cref{eq:pi_optimization}, and suppose that
$\norm{\tilde{w}_{\lambda}}_{2}>0$. Then
$a_{\lambda} = \frac{\tilde{w}_{\lambda}}{\tilde{w}_{\lambda}'w}$ solves
\cref{eq:a_optimization} with the bound given by
$B=\frac{C}{t_\lambda}\cdot {a_{\lambda}}'Z\pi_{\lambda}$. Consequently,
the worst-case bias and variance of the estimator
\begin{equation}\label{eq:optimal_estimator}
\hat{\beta}_{\lambda}={a_{\lambda}}'Y=\frac{\tilde{w}_{\lambda}'Y}{\tilde{w}_{\lambda}'w}
\end{equation}
are given by
\begin{equation}\label{eq:betahat_lambda_def}
\operatorname{\overline{bias}}_{\Gamma}(\hat\beta_\lambda)=C \overline{B}_\lambda,
\quad\text{and}\quad V_{\lambda} =
\sigma^{2}\norm{{a_{\lambda}}}_{2}^{2}, \
\quad\text{where}\;
\overline B_\lambda
=\frac{{a_{\lambda}}'Z\pi_\lambda }{\operatorname{Pen}(\pi_\lambda)} .
\end{equation}
\end{theorem}
The result follows by applying the general theory of \citet{IbKh85},
\citet{donoho94}, \citet{low95}, and \citet{ArKo18optimal} to our setting, which
allows us to rewrite \cref{eq:a_optimization} as a convex optimization problem.
Solving it then yields the result.
With the solution to \cref{eq:a_optimization} in hand, for estimation and
\ac{CI} construction, we select penalties $\lambda^{*}_{\textnormal{MSE}}$ and
$\lambda^{*}_{\textnormal{FLCI}}$ that optimize the \ac{MSE} and \ac{CI} length,
respectively. Specifically, the penalties solve the univariate optimization
problems
\begin{equation}\label{eq:optimal_lambda}
\lambda^{*}_{\textnormal{MSE}} = \operatorname*{argmin}_{\lambda} V_\lambda + (C \overline B_\lambda)^2,
\quad \lambda^{*}_{\textnormal{FLCI}}
= \operatorname*{argmin}_{\lambda} \operatorname{cv}_{\alpha}(C\overline B_\lambda/\sqrt{V_\lambda})\sqrt{V_\lambda},
\end{equation}
with $V_\lambda$ and $\overline B_\lambda$ given in
\cref{eq:betahat_lambda_def}. The optimal linear estimator is then given by
$\hat{\beta}_{\lambda^{*}_{\textnormal{MSE}}}$, and the optimal \ac{FLCI} takes
the form
$\hat{\beta}_{\lambda^{*}_{\textnormal{FLCI}}} \pm
\sigma\norm{{a_{\lambda^{*}_{\textnormal{FLCI}}}}}_{2} \cdot \operatorname{cv}_{\alpha}\left(
\frac{C}{\sigma}\frac{{a_{\lambda^{*}_{\textnormal{FLCI}}}}
'Z\pi_{\lambda^{*}_{\textnormal{FLCI}}} }{
\operatorname{Pen}(\pi_{\lambda^{*}_{\textnormal{FLCI}}})\norm{{a_{\lambda^{*}_{\textnormal{FLCI}}}}}_{2}}
\right)$.
As $t_{\lambda}\to 0$, provided that $\operatorname{Pen}(\cdot)$ is a norm on $Z_{2}$,
$\hat{\beta}_{\lambda}$ converges to the short regression estimate
$\hat{\beta}_{\textnormal{short}}=\frac{w'(I-H_{Z_{1}})Y}{w'(I-H_{Z_{1}})w}$
that only includes the unrestricted controls $Z_{1}$. This estimator minimizes
variance among all linear estimators with finite worst-case bias. In the other
direction, as $t_{\lambda}\to\infty$, $\hat{\beta}_{\lambda}$ converges to the
long regression estimate
$\hat{\beta}_{\textnormal{long}}=\frac{w'(I-H_{Z})Y}{w'(I-H_{Z})w}$, provided
that $w$ is not in the column space of $Z$ (which ensures that the condition
$\norm{\tilde{w}_{\lambda}}_{2}>0$ in~\Cref{thm:optimization_equivalence} holds
for all $\lambda$). This estimator minimizes variance among all linear
estimators that are unbiased, so \Cref{thm:optimization_equivalence} reduces to
the Gauss-Markov theorem in this case. In other words, the short and long
regressions are corner solutions of the bias-variance tradeoff, in which weight
is entirely placed on variance, or on bias.
\begin{example}[continues=example:l2]
In this case, a convenient Lagrangian formulation
for~\eqref{eq:pi_optimization} is
\begin{equation*}
\pi_{\lambda} = \operatorname*{argmin}_\pi \norm{w-Z\pi}_{2}^{2} + \lambda\norm{M\pi}_{2}^{2}.
\end{equation*}
Suppose $Z'Z + \lambda M'M$ is invertible.\footnote{Invertibility holds so
long as no element $\pi\ne 0$ satisfies $Z\pi=0$ and $M\pi=0$
simultaneously. Intuitively, if $Z$ has rank less than $k$, then the data is
not informative about certain directions $\pi$, and we require the matrix
$M$ to place sufficient restrictions on $\pi$ in these directions.} Then the
first order conditions immediately imply the closed form solution
\begin{equation*}
\pi_\lambda = (Z'Z + \lambda M'M)^{-1}Z'w,
\end{equation*}
which is a generalized ridge regression estimator of the propensity
score.\footnote{We reserve the term ``ridge regression'' without the qualifier
``generalized'' for the case $M'M=I_k$.} Plugging this expression for
$\pi_{\lambda}$ into \cref{eq:optimal_estimator} yields
\begin{equation*}
\hat\beta_\lambda
=e_1'\left(X'X+\lambda
\begin{pmatrix}
0&0\\
0&M'M
\end{pmatrix}
\right)^{-1}X'Y,
\end{equation*}
where $e_1=(1, 0, \dotsc, 0)'$ is the first standard basis vector. Thus, the
optimal estimate can also be obtained from a generalized ridge regression of
$Y$ onto $X$. The optimality of ridge regression in this setting was shown by
\citet{li82}, and the above derivation gives this result as a special case of
\Cref{thm:optimization_equivalence}. Under the \citet{li_linear_2020}
specification $M=(0,(Z_{2}'(I-H_{Z_{1}})Z_{2}/n)^{1/2})$, the estimator
further simplifies to a weighted average of the short and long regression
estimates,
\begin{equation}\label{eq:li_mueller_beta}
\hat{\beta}_{\lambda}
=\omega(\lambda)\hat{\beta}_{\textnormal{short}}
+ (1-\omega(\lambda))\hat{\beta}_{\textnormal{long}},
\end{equation}
with weights
\begin{equation*}
\omega(\lambda)=\frac{\lambda/n}{
\lambda/n+ \varsigma^{2}
},\qquad \varsigma^{2}=\frac{w'(I-H_{Z})w}{w'(I-H_{Z_{1}})w}=
\frac{\operatorname{var}(\hat{\beta}_{\textnormal{short}})}{\operatorname{var}(\hat{\beta}_{\textnormal{long}})}.
\end{equation*}
The weight on the short regression increases with $\lambda$ (as the relative
weight on variance in the bias-variance tradeoff increases), and decreases
with $\varsigma^{2}$.
\end{example}
\begin{example}[continues=example:l1]
In this case, the solution to~\eqref{eq:pi_optimization} is given by a variant
of the lasso estimate \citep{tibshirani_regression_1996} that only penalizes
$\gamma_{2}$.
The resulting estimator $\hat{\beta}_{\lambda}$ is related to estimators
proposed for constructing \acp{CI} using the lasso \citep[see, among
others,][]{zhang_confidence_2014,javanmard_confidence_2014,van_de_geer_asymptotically_2014,belloni_inference_2014}.
These papers propose estimators for $\beta$ that combine lasso estimates from
the outcome regression of $Y$ onto $X$ with lasso estimates from the
propensity score regression, which yields an estimate that is non-linear in
$Y$. In contrast, our estimator only uses lasso estimates for the propensity
score regression, and is linear in $Y$. We give a detailed comparison between
our estimator and this ``double lasso'' approach in
\Cref{sec:nonconvex_efficiency}.
While under the $\ell_{2}$ constraints with
$M=(0,(Z_{2}'(I-H_{Z_{1}})Z_{2}/n)^{1/2})$, the class of optimal estimators in
\cref{eq:li_mueller_beta} depends on the data only through the short and long
regression estimates, $\hat{\beta}_{\lambda}$ under $\ell_{1}$ constraints
(or under $\ell_2$ constraints with other choices of $M$)
doesn't simply interpolate between these two extremes, and the optimal weights
$\alpha_{\lambda}$ display much richer variation. This is analogous to the
form of optimal weights in regression discontinuity designs
\citep[e.g.][]{ArKo18optimal,ImWa19}, where one needs to consider a range of
bandwidths, rather than just interpolating between estimators that consider
the maximal and minimal possible bandwidths.
\end{example}
\begin{example}[name={Partly linear model},label={example:partly_linear_model}]
To flexibly control for a low-dimensional set of covariates $\tilde{z}_{i}$,
one may specify a semiparametric model
\begin{equation*}
y_{i} = w_{i}\beta + h(\tilde{z}_{i}) + \varepsilon_{i},
\quad \widetilde{\operatorname{Pen}}(h)\le \widetilde C,
\end{equation*}
where the penalty $\widetilde{\operatorname{Pen}}(h)$ is a seminorm on functions $h(\cdot)$
that penalizes the ``roughness'' of $h$, such as the Hölder or Sobolev
seminorm of order $q$. Minimax linear estimation in this model for particular
choices of $\widetilde{\operatorname{Pen}}(h)$ has been considered in
\citet{heckman_minimax_1988}. This setting is covered by our setup if we
define $Z=I_{n}$, $\gamma_{i}=h(\tilde{z}_{i})$, and
$\operatorname{Pen}(\gamma)=\min_{h\colon h(\tilde z_i)=\gamma_i, \; i=1,\dotsc n}
\widetilde{\operatorname{Pen}}(h)$ (assuming the minimum is taken).
\Cref{thm:optimization_equivalence} then implies that the optimal estimator
takes the form
\begin{equation*}
\hat{\beta}_\lambda = \frac{\sum_{i=1}^{n}
(w_i-g_\lambda(\tilde z_i))Y_i}{\sum_{i=1}^n(w_i-g_\lambda(\tilde z_i))w_i},
\end{equation*}
where $g_\lambda(\cdot)$ is analogous to the regularized regression estimate
$\pi_\lambda$ in~\eqref{eq:pi_optimization}: it solves
\begin{equation*}
\min_{g} \sum_{i=1}^n(w_i-g(\tilde z_i))^2
\quad\text{s.t.}\quad
\widetilde{\operatorname{Pen}}(g) \le t_\lambda.
\end{equation*}
When $\widetilde{\operatorname{Pen}}$ is the Sobolev seminorm, this yields a spline estimate
$g_{\lambda}$ \citep[see, for example][]{wahba90}. Interestingly, the
estimator proposed in the seminal work by
\citet{robinson_root-n-consistent_1988} takes a similar form to the estimator
$\hat\beta_\lambda$, involving residuals from a nonparametric regression of
$w$ on $\tilde z_i$. While the analysis in
\citet{robinson_root-n-consistent_1988} is asymptotic, our results imply that
a version of this estimator has sharp finite-sample optimality properties.
\end{example}
\subsection{Efficiency among non-linear procedures}\label{sec:linear_efficiency}
So far, we have restricted attention to procedures that are linear in the
outcomes $Y$. We now show that the estimator
$\hat{\beta}_{\lambda^{*}_{\textnormal{MSE}}}$, and the \ac{CI} based on the
estimator $\hat{\beta}_{\lambda^{*}_{\textnormal{FLCI}}}$ are in fact highly
efficient among all procedures, not just linear ones. This is due to the
convexity and symmetry of the parameter space $\Gamma$, and follows from the
general results in \citet{donoho94}, \citet{low95} and \citet{ArKo18optimal} for
estimation of linear functionals in Gaussian models with convex parameter
spaces.
\begin{corollary}\label{optimality_corollary}
Let $\lambda^{*}_{\textnormal{MSE}}$ and $\lambda^{*}_{\textnormal{FLCI}}$ be
given in~\cref{eq:optimal_lambda}, where the optimization is over all
$\lambda$ with $t_{\lambda}>0$ such that $\norm{w-Z\pi_{\lambda}}_{2}>0$.
Let $\hat\beta_\lambda$, $\overline B_\lambda$ and $V_{\lambda}$ be given in
\cref{eq:betahat_lambda_def}. Let $\tilde \beta$ and
$\tilde\beta\pm \tilde\chi$ denote some other (possibly non-linear) estimator
and some other (possibly non-linear, variable-length) \ac{CI}.
\begin{enumerate}[label=({\roman*})]
\item\label{item:bias-variance} For any $\lambda$,
$\sup_{\beta\in\mathbb{R}, \gamma\in\Gamma} \operatorname{var}_{\beta,
\gamma}(\tilde\beta) \leq V_\lambda$ implies
$\operatorname{\overline{bias}}_{\Gamma}(\tilde\beta) \geq C \overline B_\lambda$, and
$\operatorname{\overline{bias}}_{\Gamma}(\tilde\beta) \leq C \overline B_\lambda$ implies
$\sup_{\beta\in\mathbb{R}, \gamma\in\Gamma} \operatorname{var}_{\beta, \gamma}(\tilde\beta)
\geq V_\lambda$.
\item\label{item:mse} The worst-case \ac{MSE} improvement of $\tilde\beta$
over $\hat\beta_{\lambda^*_{\textnormal{MSE}}}$ is bounded by
\begin{equation*}
\frac{\ensuremath\operatorname{R}_{\textnormal{MSE}}(\tilde\beta)}{\ensuremath\operatorname{R}_{\textnormal{MSE}}(\hat\beta_{\lambda^*_{\textnormal{MSE}}})}\ge
\kappa^*_{\textnormal{MSE}}(X, \sigma, \Gamma)\ge 0.8,
\end{equation*}
where $\kappa^*_{\textnormal{MSE}}(X, \sigma, \Gamma)$ is given in
\Cref{sec:proof-coroll-optimality}.
\item\label{item:flci} The improvement of the expected length of the \ac{CI}
$\tilde\beta\pm\tilde \chi$ over the optimal linear \ac{FLCI}
$\hat\beta_{\lambda^*_{\textnormal{FLCI}}} \pm \operatorname{cv}_{\alpha}(C\overline
B_{\lambda^*_{\textnormal{FLCI}}}/V_{\lambda^*_{\textnormal{FLCI}}}^{1/2})
V_{\lambda^*_{\textnormal{FLCI}}}^{1/2}$ at $\gamma=0$ and any $\beta$ is
bounded by
\begin{equation*}
\frac{E_{\beta,0}[\tilde\chi]}{\operatorname{cv}_{\alpha}(C\overline{B}_{\lambda^*_{\textnormal{FLCI}}}/
V_{\lambda^*_{\textnormal{FLCI}}}^{1/2})V_{\lambda^*_{\textnormal{FLCI}}}^{1/2}}
\ge \kappa^*_{\textnormal{FLCI}}(X, \sigma, \Gamma),
\end{equation*}
where $\kappa^*_{\textnormal{FLCI}}(X, \sigma, \Gamma)$ is given in
\Cref{sec:proof-coroll-optimality} and is at least $0.717$ when $\alpha=0.05$.
\end{enumerate}
\end{corollary}
By construction, the estimator $\hat{\beta}_{\lambda}$ minimizes variance among
all linear estimators with a bound $C\overline{B}_{\lambda}$ on the bias (or
equivalently, it minimizes bias among all linear estimators with a bound
$V_{\lambda}$ on the variance).
\Cref{optimality_corollary}\ref{item:bias-variance} shows that this optimality
property is retained if we enlarge the class of estimators to all estimators,
including non-linear ones. As a result, the minimax linear estimator
$\hat{\beta}_{\lambda^{*}_{\textnormal{MSE}}}$ (i.e.\ the estimator attaining
the lowest worst-case \ac{MSE} in the class of linear estimators) continues to
perform well among all estimators, including non-linear ones: by
\Cref{optimality_corollary}\ref{item:mse}, its worst-case \ac{MSE} efficiency is
at least 80\%. The exact efficiency bound
$\kappa^*_{\textnormal{MSE}}(X, \sigma, \Gamma)$ depends on the design matrix,
noise level, and particular choice of the parameter space, and can be computed
explicitly in particular applications. We have found that typically the
efficiency is considerably higher than the 80\% lower bound.
Finally, \Cref{optimality_corollary}\ref{item:flci} shows that it is not
possible to substantively improve upon the \ac{FLCI} based on
$\hat{\beta}_{\lambda^{*}_{\textnormal{FLCI}}}$ in terms of expected length when
$\gamma=0$, even if we consider variable length \acp{CI} that ``direct power''
at $\gamma=0$ (potentially at the expense of longer expected length when
$\gamma\neq 0$). The construction of the \ac{FLCI} may appear conservative: its
length depends on the worst-case bias over the parameter space for
$(\beta, \gamma')'$, which, as the proof of \Cref{thm:optimization_equivalence}
shows, attains at
$\gamma=Ct_{\lambda^{*}_{\textnormal{FLCI}}}^{-1}\pi_{\lambda^{*}_{\textnormal{FLCI}}}$,
with $\operatorname{Pen}(\gamma)=C$. Therefore, one may be concerned that when the magnitude
of $\gamma$ is much smaller than $C$, the \ac{FLCI} is too long.
\Cref{optimality_corollary}\ref{item:flci} shows that this is not the case, and
the efficiency of the \ac{FLCI} is at least 71.7\% relative to variable-length
\acp{CI} that optimize their expected length when $\gamma=0$. The exact
efficiency bound $\kappa^*_{\textnormal{MSE}}(X, \sigma, \Gamma)$ can be
computed explicitly in particular applications, and we have found that it is
typically considerably higher than $71.7\%$.
A consequence of \Cref{optimality_corollary}\ref{item:flci} is that it is
impossible to form a CI that is adaptive with respect to the regularity parameter
$C$ that bounds $\operatorname{Pen}(\gamma)$. In the present setting, an adaptive CI would
have length that automatically reflects the true regularity $\operatorname{Pen}(\gamma)$ while
maintaining coverage under a conservative a priori bound on $\operatorname{Pen}(\gamma)$.
However, according to \Cref{optimality_corollary}\ref{item:flci}, any CI must
have expected width that reflects the conservative a priori bound $C$ rather
than the true regularity $\operatorname{Pen}(\gamma)$, even when $\operatorname{Pen}(\gamma)$ is much
smaller than the conservative a priori bound $C$. In particular, it is
impossible to automate the choice of the regularity parameter $C$ when forming a
CI\@. We therefore recommend varying $C$ as a form of sensitivity analysis, or
using auxiliary information to choose $C$; see \Cref{choice_of_C_remark}.
\subsection{Heterogeneous treatment effects}\label{sec:heter-treatm-effects}
If $w_{i}$ is as good as randomly assigned conditional on $z_{i}$, the
coefficient $\beta$ in \cref{linear_regression_eq} can be interpreted as the
\ac{ATE} of a one-unit increase in the variable $w$. This interpretation
requires that the individual \acp{TE} are mean independent of $z_{i}$. To relax
this assumption, we replace $\beta$ in \cref{linear_regression_eq} with a
covariate-specific coefficient $\beta(z)$ that represents the conditional
\ac{ATE} for units $i$ with $z_{i}=z$, obtaining the model
\begin{equation}\label{het_te_regression_eq}
Y_i = w_i\beta(z_i) + z_i'\gamma + \varepsilon_i.
\end{equation}
Suppose the parameter of interest takes the form $\int \beta(z)\, d\mu(w, z)$
where $\mu$ is a signed measure defined on some set that includes the empirical
support $\{(w_{i}, z_{i})\}_{i=1}^{n}$. Allowing $\mu$ to be signed allows for
inference on non-convex averages of $\beta(z)$. We allow $\mu$ to place nonzero
mass outside the empirical support $\{(w_{i}, z_{i})\}_{i=1}^{n}$, thereby
allowing for extrapolation. The general theory of estimation and inference on
linear functionals developed in \citet{donoho94} and \citet{ArKo18optimal}
underlying \Cref{thm:optimization_equivalence,optimality_corollary} can be
applied to inference on this parameter under any convex and symmetric
restriction on the function $\beta(\cdot)$ and the parameter $\gamma$---only the
worst-case bias calculation in \cref{eq:maxbias_def} changes. We now discuss two
particular specifications for $\mu$ and $\beta(\cdot)$.
For the first approach, let $\mu$ correspond to a weighted empirical measure
with weights $c_{i}$ that sum to one, so the parameter of interest is given by
$\tilde{\beta}=\sum_{i}c_{i}\beta(z_{i})$. For example, the unweighted case
$c_{i}=1/n$ gives the (conditional on the sample) \ac{ATE}, while setting
$c_{i}=w_{i}/\sum_{j}w_{j}$ gives the \ac{ATE} for the treated. Assume also that
the function $\beta(z)$ is linear, $\beta(z)=z'\delta$, and that the first
element of $z_{i}$ is a constant. Consider a parameter space for
$(\delta',\gamma')'$ given by
$\{(\delta',\gamma')'\in\mathcal{B}\colon \operatorname{Pen}(\delta,\gamma)\leq C\}$, where
$\mathcal{B}$ is a subspace of a Euclidean space with a seminorm $\operatorname{Pen}$.
Allowing $\mathcal{B}$ to be a subspace allows us to restrict $\beta(z)$ to only
depend on a subset of the controls. As noted by \citet[Section 5.3]{imbens_recent_2009} in the unpenalized case, we can map this problem back
into the model in \cref{linear_regression_eq,eq:Gamma_def} by rewriting
\cref{het_te_regression_eq} under these assumptions as
\begin{equation*}
Y_i = w_i\tilde{\beta}
+w_{i}(z_{i,-1}-\sum_{j}c_{j}z_{j,-1})'\delta_{-1} + z_{i}'\gamma + \varepsilon_i,
\end{equation*}
where $z_{i,-1}$ is the vector of controls excluding a constant and
$\delta_{-1}$ is the corresponding subvector of $\delta$. This is exactly our
problem in \cref{linear_regression_eq}, with the control vector consisting of
the original controls $z_{i}$ as well as the interaction of the treatment with
the demeaned controls, $w_{i}(z_{i,-1}-\sum_{i}c_{i}z_{i,-1})$. We use this
approach in our empirical application in \Cref{sec:empirical-app}.
For the second approach, we compute the same linear estimator and bias-aware
\acp{CI} as in the homogeneous \ac{TE} model in \cref{linear_regression_eq}; we
only change the interpretation of the estimand as targeting a particular
weighted average of \acp{TE} given in the next theorem. To set the stage for the
theorem, let us denote the worst-case bias of a linear estimator $\hat{\beta}$
relative to an estimand $\int \beta(z)d\mu(w, z)$ when the heterogeneity is
completely unrestricted by
$\widetilde{\operatorname{\overline{bias}}}_{\Gamma}(\hat{\beta};\mu)=\sup_{\beta(\cdot), \gamma}
\left[ \sum_{i=1}^n a_{i}w_i \beta(z_i)+a'Z\gamma - \int \beta(z)\, d\mu(w, z)
\right]$, where the supremum is over $\gamma\in\Gamma$ and all functions
$\beta(\cdot)$. For an $n$-vector $a$, let $\mu_{a, w}^{*}(w, z)$ denote a
weighted empirical measure with (possibly negative) weights $a_{i}w_{i}$, so
that the parameter of interest becomes
$\tilde{\beta}_{a, w}=\sum_{i}a_{i}w_{i}\beta(z_{i})$.
\begin{theorem}\label{theorem:weighted_te}
Let $\mu$ be a signed measure with $\int\, d \mu(w, z)=1$, and let
$\hat{\beta}=a'Y$ be an estimator with $a'w=1$. If $\mu=\mu^*_{a, w}$, then
$\widetilde{\overline{\operatorname{bias}}}_{\Gamma}(\hat\beta;\mu)
=\operatorname{\overline{bias}}_{\Gamma}(\hat\beta)$, with $\operatorname{\overline{bias}}_{\Gamma}(\hat\beta)$ given
in~\cref{eq:maxbias_def}. If $\mu\ne \mu^*_a$, then
$\widetilde{\overline{\operatorname{bias}}}_{\Gamma}(\hat\beta;\mu)=\infty$. Furthermore,
the estimator $\hat{\beta}_{\lambda}$ given in \cref{eq:optimal_estimator}
solves
\begin{equation}\label{moving_goalposts_eq}
\min_{\hat\beta=a'Y, a\in\mathbb{R}^{n}} \operatorname{var}(\hat\beta)
\quad\text{s.t.}\quad
\min_\mu \widetilde{\overline{\operatorname{bias}}}_{\Gamma}(\hat\beta)\le C\overline B_\lambda,
\end{equation}
where the second minimization is over all signed measures such that
$\int d\mu(w, z)=1$.
\end{theorem}
The theorem gives three results for inference when $\beta(\cdot)$ is
unrestricted. First, given a linear estimator $a'Y$, the only estimand for which
the bias is finite is $\tilde{\beta}_{a, w}$. Second, for this estimand, the
bias is the same as in the homogeneous \ac{TE} model with $\beta(z)=\beta$.
Thus, assuming homogeneous \acp{TE} leads to valid inference for
$\tilde{\beta}_{a, w}$ when \acp{TE} are in fact heterogeneous. In particular,
the bias-aware \ac{CI} based on the estimator $\hat{\beta}_{\lambda}$ provides
inference on the weighted average
$\tilde{\beta}_{a_{\lambda}, w}=
\sum_{i}\tilde{w}_{\lambda, i}w_{i}\beta(z_{i})/\sum_{i}\tilde{w}_{\lambda, i}w_{i}$.
Observe that, when the treatment $w_{i}$ is binary, the weights are non-negative
if and only if the residual in the propensity score regression
$\tilde{w}_{\lambda, i}$ is positive whenever $w_{i}=1$---this can be easily
verified in a given application. Equivalently, the weights are positive if the
fitted values $z_{i}'\pi_{\lambda}$ are smaller than one: the fitted values in
the propensity score regression must respect the population constraint that
treatment probabilities must be smaller than one. This finding is a
finite-sample analog of the identification result in \citet{ghk21}, who show
that the estimand in the partly linear model has an analogous weighted \ac{ATE}
interpretation under heterogeneous \acp{TE}. An analogous identification result
in a random design setting dates to at least
\citet{angrist_estimating_1998}, who gave a weighted \ac{ATE} interpretation to
the \ac{OLS} estimand.
Third, the estimator $\hat{\beta}_{\lambda}$ remains optimal in the
heterogeneous \ac{TE} model in that it solves a bias-variance tradeoff in a
problem where we can pick the estimand to make the estimation problem as easy as
possible. The problem~\eqref{moving_goalposts_eq} is a finite-sample version of
the ``moving the goalposts'' problem considered by \citet[][Section
5.4]{crump_moving_2006}.\footnote{The optimization
problem~\eqref{moving_goalposts_eq} was also used by \citet{ImWa19} in the
context of a regression discontinuity design with multiple cutoffs, although
they did not explicitly note the optimality properties of the resulting
estimator.} \citet{crump_moving_2006} derived the measure $\mu$ that minimizes
the asymptotic variance of a particular class of inverse propensity score
weighted estimators of weighted \acp{ATE} in a random design setup.
\citet{ghk21} show this measure in fact minimizes the variance among all regular
estimators; the measure coincides with the weighting of the treatment effects
given in \citet{angrist_estimating_1998} under homoskedasticity, giving an
optimality property to the \ac{OLS} estimator.
\section{Implementation with non-Gaussian and heteroske\-dastic errors}\label{sec:impl-with-non}
We now discuss practical implementation issues, allowing $\varepsilon$ to be
non-Gaussian and heteroskedastic. As a baseline, we propose the following
implementation:
\begin{algorithm}[Baseline implementation]\mbox{}\label{algorithm:baseline}
\begin{description}
\item[Input] Data $(Y, X)$, penalty $\operatorname{Pen}(\cdot)$, regularity parameter $C$,
and initial estimates of residuals
$\hat{\varepsilon}_{\textnormal{init}, 1}, \dotsc,
\hat{\varepsilon}_{\textnormal{init}, n}$.
\item[Output] Estimator and \ac{CI} for $\beta$.
\end{description}
\begin{enumerate}
\item\label{initial_variance_estimator_step} Compute an initial variance estimator,
$\hat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}
\hat{\varepsilon}_{\textnormal{init}, i}^2$, assuming homoskedasticity.
\item\label{solution_path_step} Compute the solution path
$\{\pi_{\lambda}\}_{\lambda>0}$ for the regularized propensity score
regression in \cref{eq:pi_optimization}, indexed by the penalty weight
$\lambda$. For each $\lambda$, compute $\hat{\beta}_{\lambda}$ as in
\cref{eq:optimal_estimator}, and $\overline{B}_{\lambda}$, and $V_{\lambda}$
as in \cref{eq:betahat_lambda_def}, with $\hat{\sigma}^{2}$ in place of
$\sigma^{2}$ in the formula for $V_\lambda$.
\item Compute $\lambda^*_{\textnormal{MSE}}$ and
$\lambda^*_{\textnormal{FLCI}}$ as in~\cref{eq:optimal_lambda}, and compute
the robust variance estimate
$\hat V_{\lambda, \textnormal{rob}} = \sum_{i=1}^n {a_{\lambda, i}}^2
\hat\varepsilon_{\textnormal{init}, i}^2$, where
$a_\lambda=\frac{w-Z\pi_\lambda}{(w-Z\pi_\lambda)'w}$.
\end{enumerate}
Return the estimator $\hat\beta_{\lambda^*_{\textnormal{MSE}}}$ and the
\ac{CI}
$\hat\beta_{\lambda^*_{\textnormal{FLCI}}} \pm \operatorname{cv}_{\alpha}\left(C\overline
B_{\lambda^*_{\textnormal{FLCI}}}/\hat{V}_{\lambda^*_{\textnormal{FLCI}},
\textnormal{rob}}^{1/2}\right)\cdot
\hat{V}^{1/2}_{\lambda^*_{\textnormal{FLCI}}, \textnormal{rob}}$.
\end{algorithm}
The following remarks discuss the implementation choices, and the optimality and
validity of the baseline procedure.
\begin{remark}[Validity]\label{lindeberg_remark}
As the initial residual estimates $\hat\varepsilon_{\textnormal{init}, i}$, we
can take residuals from a regularized outcome regression of $Y$ on $X$ (see
\cref{eq:YX_regularized_regression} in \Cref{sec:global_estimation_algebra}).
We give conditions for asymptotic validity of the resulting \acp{CI} in
\Cref{sec:se_consistency}. The key requirement is that the maximal Lindeberg
weight
$\operatorname{Lind}(a_\lambda)=\max_{1\le i\le n}{a_{\lambda,
i}}^2/\sum_{j=1}^n{a_{\lambda, j}}^2$ associated with the estimator
$\hat{\beta}_{\lambda}$ shrink quickly enough relative to error in the
estimator used to form the residuals. Ensuring that
$\operatorname{Lind}(a_\lambda)$ is small prevents the estimator from putting
too much weight on a particular observation, so that the Lindeberg condition
for the central limit theorem holds.
Whether these conditions hold for the optimal estimator will in general depend
on the form of $\operatorname{Pen}(\gamma)$ and on the magnitude of $C$ relative to $n$. To
ensure that $\textnormal{Lind}(a_\lambda)$ is small enough in a particular
sample for a normal approximation to work well, one may impose a bound on this
term by only minimizing~\cref{eq:optimal_lambda} over $\lambda$ such that
$\textnormal{Lind}(a_\lambda)$ is small enough when computing
$\lambda^*_{\textnormal{FLCI}}$. This is similar to proposals by
\citet{noack_bias-aware_2019}, and \citet{javanmard_confidence_2014} in other
settings. As discussed further in \Cref{sec:se_consistency}, under mild
regularity conditions, imposing such a bound doesn't affect the convergence
rate of the resulting \ac{CI}.
\end{remark}
\begin{remark}[Efficiency]
The weights $a_{\lambda^*_{\textnormal{FLCI}}}$ and
$a_{\lambda^*_{\textnormal{MSE}}}$ are not optimal under
heteroskedasticity. One could in principle generalize the \ac{FGLS} approach
used for unconstrained estimation by deriving optimal weights under the
assumption $\varepsilon\sim \mathcal{N}(0,\Sigma)$ (which simply follows the above
analysis after pre-multiplying by $\Sigma^{-1/2}$), and derive conditions
under which the estimator and \ac{CI} that plug in an estimate of $\Sigma$ are
optimal asymptotically when the assumption of known variance and Gaussian
errors is dropped. Instead of pursuing this generalization, our baseline
implementation computes the weights $a_{\lambda}$ under the assumption of
homoskedasticity, but we use robust standard errors when computing the
\ac{CI}. Thus, analogous to the ubiquitous practice of reporting \ac{OLS} with
\ac{EHW} standard errors in the unconstrained setting, our baseline
implementation leverages homoskedasticity for efficiency, but the \acp{CI}
remain valid when the homoskedasticity assumption is violated.
\end{remark}
\begin{remark}[Choice of $C$]\label{choice_of_C_remark}
By \Cref{optimality_corollary}\ref{item:flci}, one cannot use a data-driven
rule to automate the choice of $C$ when forming a CI\@. Therefore, plausible
magnitudes of $\operatorname{Pen}(\gamma)$ need to be assessed using prior knowledge.
Such assessments can be aided by relating the magnitude of $\operatorname{Pen}(\gamma)$ to
other quantities. Let us now describe an approach to calibrating $C$ that we
use in our numerical and empirical work in
\Cref{sec:simulation-results,sec:empirical-app}. Let
$z_{i1}=(1,\tilde{z}_{i1}')'$ denote a vector of baseline controls, believed
to be important confounders, and let $z_{i2}$ be a possibly high-dimensional
vector of additional controls, believed to be less important. Suppose that
$\operatorname{Pen}(\gamma)=\norm{\gamma_{2}}$ corresponds to some norm on the additional
controls as in \Cref{example:l2,example:l1}. To formalize the belief that the
baseline controls are more important, we use the norm of the population
coefficient $\widetilde{\gamma}_{short}$ on $\tilde z_{i1}$ in the short
regression of $Y_{i}$ on a constant, $w_{i}$ and $\tilde{z}_{i1}$ as a bound
on $\norm{\gamma_{2}}$. Since $\widetilde{\gamma}_{short}$ is unknown, we set
$C^{rot}=\norm{\widehat{\gamma}_{short}}$ as a rule of thumb, where
$\widehat{\gamma}_{short}$ is an \ac{OLS} estimate of
$\widetilde{\gamma}_{short}$.\footnote{Formally, one should account for
sampling uncertainty in $\widehat{\tilde \gamma}_{short}$ to ensure validity
of the \ac{CI} under the assumption
$\norm{\gamma_{2}}\leq\norm{\tilde{\gamma}_{short}}$, such as by combining a
first stage CI for $\tilde{\gamma}_{short}$ with a Bonferroni correction. In
our Monte Carlos in \Cref{sec:simulation-results}, however, we find that the
$C^{rot}$ leads to valid coverage when this assumption holds even without
additional corrections for sampling uncertainty.}
Calibrations of the regularity parameter $C$ should be complemented by varying
$C$ as a form of sensitivity analysis. Robustness of the results can also be
assessed by computing two additional values of the regularity parameter. The
first is a ``breakdown value'' $C^*$, the largest value of $C$ such the
empirical finding of interest holds. Second, by way of a specification check,
one can form a lower \ac{CI} $\hor{\hat{\underline C}, \infty}$ for $C$ to
assess the plausibility of a given bound on $\operatorname{Pen}(\gamma)$. We present such a
\ac{CI} in \Cref{sec:C_lower_CI} for the case where $\operatorname{Pen}(\gamma)$ takes the
form of an $\ell_p$ constraint.
\end{remark}
\begin{remark}[Computational issues]
Step~\ref{solution_path_step} involves computing the solution path of a
regularized regression estimator. Efficient algorithms exist for computing
these paths under $\ell_{1}$ penalties and its variants
\citep{efron2004lars,rosset_piecewise_2007}. Under $\ell_{2}$ penalty, the
regularized regression has a closed form, so that our algorithm can again be
implemented in a computationally efficient manner. For other types of
penalties, the convexity of the optimization problem in \cref{eq:pi_optimization}
can be exploited to yield efficient implementation. We also note that since
the solution path $\pi_{\lambda}$ does not depend on $C$, it only needs to
be computed once, even when multiple choices of $C$ are considered in a
sensitivity analysis.
\end{remark}
\section{Rates of convergence}\label{sec:asymptotics}
We now derive the rates of convergence for the optimal linear \acp{FLCI} as
$n\to\infty$. For ease of notation, we assume all coefficients are constrained,
and focus on the case $\operatorname{Pen}(\gamma)=\norm{\gamma}_{p}$ for some $p\ge 1$, and
the case $\operatorname{Pen}(\gamma)=\norm{Z\gamma/\sqrt{n}}_{2}$ (see \Cref{example:l2}). We
allow the regularity parameter $C=C_{n}$ go to $0$ or $\infty$ with the sample
size, and consider high dimensional asymptotics where $k=k_n\gg n$. We consider
a standard ``high dimensional'' setting, placing conditions on the design matrix
$X$ that hold with high probability when $w_i, z_i$ are drawn i.i.d.\ over $i$,
with the eigenvalues of $\operatorname{var}((w_i, z_i')')$ bounded away from zero and infinity.
Let $q\in[0,\infty]$ denote the Hölder conjugate of $p$, satisfying $1/p+1/q=1$.
We will show that when $\operatorname{Pen}(\gamma)=\norm{\gamma}_{p}$, the optimal linear
\ac{FLCI} shrinks at the rate
\begin{equation}\label{eq:convergence-rates}
n^{-1/2}+C r_{q}(k, n)\quad\text{where}\quad
r_{q}(k, n) =
\begin{cases}
k^{1/q}/\sqrt{n} & \text{if $q<\infty$,} \\
\sqrt{\log k}/\sqrt{n} & \text{if $q=\infty$}.
\end{cases}.
\end{equation}
Furthermore, for $p=1$ and $p=2$, we will show that no other \ac{CI} can shrink
at a faster rate. For $p=1$, we will in fact prove a stronger result showing
that imposing sparsity bounds on the outcome and propensity score regressions,
in addition to the bound on $\operatorname{Pen}(\gamma)$, does not help achieve a faster rate,
unless one assumes sparsity of order greater than $C_n\sqrt{n/\log(k)}$ (termed
the ``ultra sparse'' case in \citet{cai_lasso_2017}). For the case
$\operatorname{Pen}(\gamma)=\norm{Z\gamma/\sqrt{n}}_{2}$, we will show that the optimal rate
is given by $n^{-1/2}+C$ when $k\gg n$.
If $k\gg n$ and $C=C_{n}$ does not decrease to zero with $n$, these rates
require $p<2$ (so that $q>2$) for consistency. When $p=1$, we can then allow $k$
to grow exponentially with $n$, whereas setting $1<p<2$ allows for $k$ to grow
at a polynomial rate in $n$ that depends on $p$. Since taking $C_n\to 0$ rules
out even a single coefficient being bounded away from zero, these bounds imply
that taking $p<2$ in ``high dimensional'' settings is necessary for consistency,
with $p=1$ offering the best rate conditions. It also follows from these rate
results that if $C_{n}=C$ does not decrease to zero with $n$, the bias term can
dominate asymptotically, making it necessary to explicitly account for bias in
\ac{CI} construction even in large samples.
\subsection{Upper bounds}\label{sec:asymptotic_upper_bounds}
To state the result, given $\eta>0$, let $\mathcal{E}_n(\eta)$ denote the set of
design matrices $X$ for which there exists $\delta\in\mathbb{R}^k$ such that
\begin{align*}
\frac{1}{n}\norm{w-Z\delta}_{2}^{2}&\leq \frac{1}{\eta},
& \frac{1}{n}w'(w-Z\delta)&\geq \eta,
& \frac{1}{n}\norm{Z'(w-Z\delta)}_{q}
& \le \frac{r_{q}(k, n)}{\eta}.
\end{align*}
Let
$R^{*}_{\textnormal{FLCI}}(X, C)=2\operatorname{cv}_\alpha(C
\overline{B}_{\lambda^{*}_{\textnormal{FLCI}}}/V_{\lambda^{*}_\textnormal{FLCI}}^{1/2})\cdot
V_{\lambda^{*}_\textnormal{FLCI}}^{1/2}$ denote the length of the optimal linear
\ac{FLCI}.
\begin{theorem}\label{thm:upper_rate_bound}
(i) Suppose $\operatorname{Pen}(\gamma)=\norm{\gamma}_{p}$. There exists a finite constant
$K_\eta$ depending only on $\eta$ such that
$R^{*}_{\textnormal{FLCI}}(X, C)\le K_\eta n^{-1/2}(1+ C k^{1/q})$ for $p>1$,
and $R^{*}_{\textnormal{FLCI}}(X, C)\le K_\eta n^{-1/2}(1+ C \sqrt{\log k})$
for $p=1$ for any $X\in \mathcal{E}_n(\eta)$. (ii) Suppose
$\operatorname{Pen}(\gamma)=\norm{Z\gamma/\sqrt{n}}_{2}$. There exists a finite constant
$K_\eta$ depending only on $\eta$ such that
$R^{*}_{\textnormal{FLCI}}(X, C)\le K_\eta (n^{-1/2}+ C)$ for any $X$ such
that $\eta\leq w'w/n$.
\end{theorem}
The second part of the \namecref{thm:upper_rate_bound} follows since the short
regression without any controls achieves a bias that is of the order $C$. The
first part shows that the upper bounds on the rate of convergence match those in
\cref{eq:convergence-rates} if the high-level condition
$X\in \mathcal{E}_n(\eta)$ holds. The next lemma shows that this high-level
condition holds with high probability when $w_{i}, Z_{i}$ are drawn i.i.d.\ from
a distribution satisfying mild conditions on moments and covariances.
\begin{lemma}\label{upper_bound_event_lemma}
Suppose $w_i, z_i$ are drawn i.i.d.\ over $i$, and let
$\delta=\operatorname*{argmin}_b E[(w_i-z_i'b)^2]$ so that $z_{i}'\delta$ is the population
best linear predictor error of $w_{i}$. Suppose that the linear prediction
error $E[(w_{i}-z_{i}'\delta)^{2}]$ is bounded away from zero as
$k\to \infty$, $E[w_i^2]<\infty$, and that
$\sup_j E[\abs{(w_{i}-z_{i}'\delta)z_{ij}}^{\max\{2,q\}}]<\infty$ when $p>1$, and, for some
$c>0$, $P\left(\abs{(w_{i}-z_{i}'\delta)z_{ij}}\ge t \right)\le 2\exp(-c t^2)$ for all $j$
when $p=1$. Then, for any $\tilde\eta>0$, there exists $\eta$ such that
$X\in \mathcal{E}_n(\eta)$ with probability at least $1-\tilde\eta$ for large
enough $n$.
\end{lemma}
\subsection{Lower bounds}
We now show that the rates in~\cref{eq:convergence-rates} are sharp when
$p=2$, or $p=1$.
\subsubsection{\texorpdfstring{$p=2$}{p=2}}\label{sec:l2_lower_bounds}
As with the upper bound in \Cref{sec:asymptotic_upper_bounds}, we derive a bound
that holds when the design matrix $X$ is in some set, and then show that this
set has high probability when $w_i, z_i$ are drawn i.i.d.\ from a sequence of
distributions satisfying certain conditions. We focus on the case $k\geq n$. Let
$\widetilde{\mathcal{E}}_n(\eta)$ denote the set of design matrices $X$ such
that
\begin{equation*}
\eta\le \frac{1}{n}w'w \le \eta^{-1},
\quad \min\operatorname{eig}(ZZ'/k)\ge \eta,
\end{equation*}
where $\operatorname{eig}(A)$ denotes the set of eigenvalues of a square matrix $A$.
\begin{theorem}\label{thm:l2_lower_bound}
Let $\hat\beta\pm\hat\chi$ be a \ac{CI} with coverage at least $1-\alpha$
under $\operatorname{Pen}(\gamma)\le C$. (i) If $\operatorname{Pen}(\gamma)=\norm{\gamma}_{2}$, there
exists a constant $c_\eta>0$ depending only on $\eta$ such that the expected
length under $\beta=0$, $\gamma=0$ satisfies
$E_{0, 0}[\hat \chi]\ge c_\eta n^{-1/2}(1+ C k^{1/2})$ for any
$X\in \widetilde{\mathcal{E}}_{n}(\eta)$. (ii) If
$\operatorname{Pen}(\gamma)=\norm{Z\gamma/\sqrt{n}}_{2}$, there exists a constant $c_\eta>0$
depending only on $\eta$ such that the expected length under $\beta=0$,
$\gamma=0$ satisfies $E_{0, 0}[\hat \chi]\ge c_\eta (n^{-1/2}+ C)$
for any $X\in \widetilde{\mathcal{E}}_{n}(\eta)$.
\end{theorem}
If $z_{i}$ is i.i.d.\ over $i$, then $EZZ'/k $ is equal to the $n\times n$
identity matrix times the scalar $\frac{1}{k}\sum_{j=1}^k E[z_{ij}^2]$. Thus,
the condition on the minimum eigenvalue of $ZZ'/k$ will hold under concentration
conditions on the matrix $Z'Z$ so long as the second moments of the covariates
are bounded from below. Here, we state a result for a special case where the
$z_{ij}$'s are i.i.d.\ normal, which is immediate from \citet[][Lemma
3.4]{donoho_for_2006}.
\begin{lemma}
Suppose that $w_{i}$ are i.i.d.\ over $i$ and that $z_{ij}$ are i.i.d.\ normal over $i$
and $j$. Then, for any $\tilde\eta>0$, there exists $\eta>0$ such that $X\in
\widetilde{\mathcal{E}}_n(\eta)$ with probability at least $1-\tilde\eta$ once $n$
and $k/n$ are large enough.
\end{lemma}
\subsubsection{\texorpdfstring{$p=1$}{p=1}}
We now consider the case where $p=1$, as in \Cref{example:l1}. Rather than
imposing conditions on $X$ in a fixed design setting that hold with high
probability (as in \Cref{sec:asymptotic_upper_bounds} and
\Cref{sec:l2_lower_bounds}), we directly consider a random design setting, and
we do not condition on $X$ when requiring coverage of \acp{CI}. This allows us
to strengthen the conclusion of our theorem by showing that the rate in
\Cref{thm:upper_rate_bound} is sharp even if one imposes a linear model for
$w_i$ given $z_i$ along with sparsity and $\ell_1$ bounds on the coefficients in
this model.
We introduce some additional notation to cover the random design setting, which
we use only in this section. We consider a random design model
\begin{align*}
Y&=w\beta+Z\gamma + \varepsilon, \quad \varepsilon\mid Z, w \sim \mathcal{N}(0,\sigma^2I_n), \\
w&=Z\delta + v, \quad v\mid Z \sim \mathcal{N}(0,\sigma_{v}^2I_n), \\
z_{ij} &\sim \mathcal{N}(0,1)\quad\text{i.i.d.\ over $i, j$}.
\end{align*}
We use $P_{\vartheta}$ and $E_{\vartheta}$ for probability and expectation when
$Y, X$ follow this model with parameters
$\vartheta=(\beta, \gamma', \delta', \sigma^2,\sigma^2_{v})'$. Let
$\sigma_0^2>0$ and $\sigma^2_{v, 0}>0$ be given and let $\Theta(C, s, \eta)$
denote the set of parameters
$\vartheta=(\beta, \gamma', \delta', \sigma^2, \sigma^2_{v})$ where
$\abs{\sigma^2-\sigma^2_0}\le \eta$,
$\abs{\sigma_{v}^2-\sigma^2_{v, 0}}\le \eta$, $\norm{\gamma}_{1}\le C$,
$\norm{\delta}_{1}\le C$, $\norm{\gamma}_0\le s$ and $\norm{\delta}_0\le s$.
\begin{theorem}\label{thm:l1_lower_bound}
Let $\hat\beta\pm \hat\chi$ be a \ac{CI} satisfying
$P_{\vartheta}(\beta\in \{\hat\beta \pm \hat{\chi}\}) \geq 1-\alpha$ for all $\vartheta$ in
$\Theta(C_n, C_n\cdot K \sqrt{n/\log k}, \eta_n)$ where $\alpha<1/2$. Suppose
$k\to\infty$, $C_n\sqrt{\log k}/n\to 0$ and
$C_n\le \sqrt{k/n}\cdot k^{-\tilde \eta}$ for some $\tilde\eta>0$. Then, there
exists $c$ such that, if $K$ is large enough and $\eta_n\to 0$ slowly enough,
the expected length of this \ac{CI} under the parameter vector $\vartheta^*$
given by $\beta=0$, $\gamma=0$, $\delta=0$, $\sigma^2=\sigma^2_0$,
$\sigma^2_{v}=\sigma^2_{v, 0}$ satisfies
$E_{\vartheta^*}[\hat\chi] \ge c\cdot n^{-1/2}(1+C_n\sqrt{\log k})$ once $n$ is
large enough.
\end{theorem}
\Cref{thm:l1_lower_bound} follows from similar arguments to
\citet{cai_lasso_2017} and \citet{javanmard_debiasing_2018}, who provide similar
bounds for the case where only a sparsity bound is imposed. According to
\Cref{thm:l1_lower_bound}, imposing sparsity does not allow one to improve upon
the \acp{CI} that uses only the $\ell_1$ bound $\norm{\gamma}_1\le C_n$ (thereby
attaining the rate in \Cref{thm:upper_rate_bound}), unless one imposes sparsity
of order greater than $C_n\sqrt{n/\log k}$. We provide further comparison with
\acp{CI} that impose sparsity in the next section.
\section{Comparison with sparsity constraints}\label{sec:nonconvex_efficiency}
Several authors have considered \acp{CI} for $\beta$ using ``double lasso''
estimators \citep[see, among
others,][]{belloni_inference_2014,javanmard_confidence_2014,van_de_geer_asymptotically_2014,zhang_confidence_2014}.
These \acp{CI} are valid under the parameter space
\begin{equation}\label{eq:sparsity_parameter_space}
\widetilde{\Gamma}(s) = \{\gamma\colon\norm{\gamma}_0\le s\},
\end{equation}
where $\norm{\gamma}_0=\#\{j\colon \gamma_j\ne 0\}$ is the $\ell_0$ ``norm,''
which indexes the sparsity of $\gamma$, and with $s$ increasing slowly enough
relative to $n$ and $k$. Since $\norm{\gamma}_0$ is not a true norm or seminorm
(it is non-convex), this parameter space is not covered by our setup.
Nonetheless, as we show in \Cref{connection_with_l1_estimator_sec}, if the
sparsity assumption is used to bound the $\ell_{1}$ loss of a preliminary lasso
estimator, arguments from \Cref{sec:finite-sample-result} lead to estimators and
\acp{CI} that are analogous to those proposed in the double lasso literature. In
\Cref{double_lasso_comparison_sec}, we provide a comparison of our approach to
these double lasso \acp{CI}.
\subsection{Connection between double lasso and optimal estimator
under \texorpdfstring{$\ell_1$}{ell1} constraints}\label{connection_with_l1_estimator_sec}
When $\operatorname{Pen}(\gamma)=\norm{\gamma}_1$ (\cref{example:l1}), the solution
$\pi_\lambda$ to \cref{eq:pi_optimization} is the lasso estimate in the
propensity score regression of $w$ on $Z$, and our
estimator~\eqref{eq:optimal_estimator} uses residuals from this lasso
regression. This is related to ``double lasso'' estimators used to form \acp{CI}
for $\beta$ under sparsity constraints on $\gamma$ \citep[see, among
others,][]{belloni_inference_2014,javanmard_confidence_2014,van_de_geer_asymptotically_2014,zhang_confidence_2014}.
For concreteness, we focus on the estimator in \citet{zhang_confidence_2014},
which is given by
\begin{equation*}
\hat\beta_{\textnormal{ZZ}} =\hat\beta_{\textnormal{lasso}}
+\frac{\tilde{w}_{\lambda}'(Y-w\hat\beta_{\textnormal{lasso}}
-Z\hat\gamma_{\textnormal{lasso}})}{\tilde{w}_{\lambda}'w},
\end{equation*}
where $\hat\beta_{\textnormal{lasso}}, \hat\gamma_{\textnormal{lasso}}$ are the lasso
estimates from regressing $Y$ on $X$:
\begin{equation*}
\hat\beta_{\textnormal{lasso}}, \hat\gamma_{\textnormal{lasso}}
= \operatorname*{argmin}_{\beta, \gamma} \norm{Y-w\beta-Z\gamma}_{2}^{2} + \tilde\lambda (\abs{\beta}
+\norm{\gamma}_{1})
\end{equation*}
for some penalty parameter $\tilde \lambda>0$.
\begin{remark}\label{nonlinearity_remark}
Note that $\hat\beta_{\textnormal{ZZ}}$ is non-linear in $Y$, due to
nonlinearity of the lasso estimates
$\hat\beta_{\textnormal{lasso}}, \hat\gamma_{\textnormal{lasso}}$, which is
consistent with the goal of efficiency in the non-convex parameter
space~\eqref{eq:sparsity_parameter_space}. In contrast,
\Cref{optimality_corollary} shows that under the convex parameter space
$\Gamma=\{\gamma\colon \norm{\gamma}_{1}\le C\}$, the estimator
$\hat\beta_{\lambda}$ in~\eqref{eq:optimal_estimator} which only uses lasso in
the propensity score regression of $w$ on $Z$, is already highly efficient
among all estimators, so that there is no further role for substantive
efficiency gains from the lasso regression of $Y$ on $X$, or from the use of
other non-linear estimators.
\end{remark}
To further understand the connection between these estimators, we note that
\citet{zhang_confidence_2014} motivate their approach by bounds of the form
\begin{equation}\label{eq:lasso_gamma_l1_bound}
\norm{\hat\gamma_{\textnormal{lasso}}-\gamma}_1 \le \tilde C\quad
\text{where}\quad\tilde C= \text{const.} \cdot s\sqrt{\log k}/\sqrt{n},
\end{equation}
which hold with high probability with the constant depending on certain ``compatibility
constants'' that describe the regularity of the design matrix $X$
\citep[see][Theorem 6.1, and references in the surrounding
discussion]{buhlmann_statistics_2011}. This suggests correcting the initial
estimate $\hat\beta_{\textnormal{lasso}}$ by estimating
$\tilde\beta=\beta-\hat\beta_{\textnormal{lasso}}$ in the regression
\begin{equation*}
\tilde Y= w(\beta-\hat\beta_{\textnormal{lasso}})+Z(\gamma-\hat\gamma_{\textnormal{lasso}})+\varepsilon
= w\tilde \beta + Z\tilde \gamma + \varepsilon,
\end{equation*}
where
$\tilde Y=Y-\hat\beta_{\textnormal{lasso}}-Z\hat\gamma_{\textnormal{lasso}}$.
Heuristically, we can treat the bound in \cref{eq:lasso_gamma_l1_bound} as a
constraint $\norm{\tilde\gamma}_{1}\le \tilde C$ on the unknown parameter
$\tilde\gamma=\gamma-\hat\gamma_{\textnormal{lasso}}$ and search for an optimal
estimator of $\tilde\beta=\beta-\hat\beta_{\textnormal{lasso}}$ under this
constraint. Applying the optimal estimator derived in
\Cref{thm:optimization_equivalence} then suggests estimating
$\beta-\hat\beta_{\textnormal{lasso}}$ with
$\frac{\tilde{w}_{\lambda}'\tilde Y}{\tilde{w}_{\lambda}'w}$. Adding this
estimate to $\hat\beta_{\textnormal{lasso}}$ gives the estimate
$\hat\beta_{\textnormal{ZZ}}$ proposed by \citet{zhang_confidence_2014}. Whereas
\citet{zhang_confidence_2014} motivate their approach as one possible way of
correcting the initial estimate $\hat\beta_{\textnormal{lasso}}$ using the bound
in \cref{eq:lasso_gamma_l1_bound}, the above analysis shows that their
correction is in fact identical to an approach in which one optimizes this
correction numerically.\footnote{The estimator proposed by
\citet{javanmard_confidence_2014} performs a numerical optimization of this
form, but with the constraint~\eqref{eq:lasso_gamma_l1_bound} replaced by a
constraint on
$\abs{\hat\beta_{\textnormal{lasso}}-\beta}+\norm{\hat\gamma_{\textnormal{lasso}}-\gamma}_{1}$.
Thus, \Cref{thm:optimization_equivalence} shows that a modification of the
constraint used in \citet{javanmard_confidence_2014} yields the same estimator
as \citet{zhang_confidence_2014}.}
Under the bound in \cref{eq:lasso_gamma_l1_bound} it follows that
$\hat\beta_{\textnormal{ZZ}}-\beta =\tilde b + {a_\lambda}'\varepsilon$ where
$a_\lambda=\frac{\tilde{w}_\lambda}{\tilde{w}_{\lambda}'w}$ are the optimal weights
under the $\ell_1$ constraint $\norm{\tilde\gamma}_{1}\le \tilde C$, given in
\Cref{thm:optimization_equivalence}. Furthermore,
$\abs{\tilde b}\le \tilde C\overline B_\lambda$, with $\overline B_\lambda$
given in \Cref{thm:optimization_equivalence} and $\tilde C$ given in
\cref{eq:lasso_gamma_l1_bound}, and the variance of the random term
${a_\lambda}'\varepsilon$ is given by $V_\lambda$ in
\Cref{thm:optimization_equivalence}. Using arguments similar to those used to
prove \Cref{thm:upper_rate_bound}, it follows that
$\tilde C\overline B_\lambda/\sqrt{V_\lambda}$ is bounded by a constant times
$s(\log k)/\sqrt{n}$, so that one can ignore bias in large samples as long as
this term converges to zero. This leads to the \ac{CI} proposed by
\citet{zhang_confidence_2014}, which takes the form
\begin{equation}\label{eq:zz_CI}
\{\hat{\beta}_{\textnormal{ZZ}} \pm z_{1-\alpha/2} \hat{V}_\lambda^{1/2}\},
\end{equation}
where $\hat V_\lambda$ is an estimate of the variance $V_\lambda$. We use the
term ``double lasso \ac{CI}'' to refer to this \ac{CI}, and to related \acp{CI}
such as those proposed in
\citet{belloni_inference_2014,javanmard_confidence_2014,van_de_geer_asymptotically_2014}.
\begin{remark}\label{bias-aware_double_lasso_remark}
To avoid the assumption that $s(\log k)/\sqrt{n}\to 0$ one could, in
principle, extend our approach and the above analysis to form valid bias-aware
\acp{CI} as
$\{\hat\beta_{\textnormal{ZZ}} \pm [ \tilde{C}
\overline{B}_\lambda+z_{1-\alpha/2}\hat{V}_\lambda^{1/2}] \}$.\footnote{We use
the slightly more conservative approach of adding and subtracting the bound
$\tilde {C}\overline{B}_\lambda$ rather than using the critical value
$\operatorname{cv}_{\alpha}(\tilde {C}\overline{B}_\lambda/\hat{V}_\lambda^{1/2})$ as in
\cref{eq:linear-CI}, since the ``bias'' term for
$\hat\beta_{\textnormal{ZZ}}$ is correlated with $\varepsilon$ through the
first step estimates
$\hat\beta_{\textnormal{lasso}}, \hat\gamma_{\textnormal{lasso}}$.}
Unfortunately, finding a computable constant $\tilde C$
in~\eqref{eq:lasso_gamma_l1_bound} that is sharp enough to yield useful bounds
in practice appears to be difficult, although it is an interesting area for
future research.
\end{remark}
\subsection{Comparison of our approach with CIs based on double lasso
estimators}\label{double_lasso_comparison_sec}
When should one use a double lasso \ac{CI}, and when should one use the
approach in the present paper? In principle, this depends on the a priori
assumptions one is willing to make, and whether they are best captured by a
sparsity bound or a bound on convex penalty function, such as the $\ell_{1}$ or
$\ell_{2}$ norm. In many settings, it may be difficult to motivate the assumption
that a regression function has a sparse approximation, whereas upper bounds on
the magnitude of the coefficients may be more plausible.
A key advantage of the \acp{CI} and estimators we propose is that they have
sharp finite-sample optimality properties and coverage guarantees in the fixed
design Gaussian model with known error variance. While this is an idealized
setting, the worst-case bias calculations do not depend on the error
distribution, and remain the same under non-Gaussian, heteroskedastic errors.
Our approach directly accounts for the potential finite-sample bias of the
estimator, rather than relying on ``asymptotic promises'' about rates at which
certain constants involved in bias terms converge to zero.
On the flip side, our \acp{CI} require an explicit choice of the regularity
parameter $C$ in order to form a ``bias-aware'' \ac{CI}. In contrast, \acp{CI}
based on double lasso estimators do not require explicitly choosing the
regularity (in this case, the sparsity $s$), since they ignore bias. This is
justified under asymptotics in which $s$ increases more slowly than
$\sqrt{n}/\log k$, which lead to the bias of $\hat\beta_{\textnormal{ZZ}}$
decreasing more quickly than its standard deviation. Thus, the \ac{CI} in
\cref{eq:zz_CI} is ``asymptotically valid'' without the need to explicitly
specify the sparsity index $s$: one need only make an ``asymptotic promise''
that $s$ increases slowly enough. However, such asymptotic promises are
difficult to evaluate in a given finite-sample setting. Indeed, as shown by
\citet{wuthrich_omitted_2021} and confirmed in our Monte Carlos in
\Cref{sec:simulation-results} below, the double lasso CI leads to undercoverage
in finite samples even in relatively sparse settings. To ensure good
finite-sample coverage of the \ac{CI} in~\cref{eq:zz_CI}, one needs to ensure
that the actual finite-sample bias is negligible relative to the standard
deviation of the estimator. But since any bias bound depends on the sparsity
index $s$ (as in the bound in \cref{eq:lasso_gamma_l1_bound}), this gets us back
to having to explicitly specify $s$.
Thus, \acp{CI} that ignore bias such as conventional \acp{CI} based on double
lasso estimators do not avoid the problem of specifying $s$ or $C$: they merely
make such choices implicit in their asymptotic promises. These issues show up
formally in the asymptotic analysis of such \acp{CI}. In particular, double
lasso \acp{CI} require the ``ultra sparse'' asymptotic regime
$s=o(\sqrt{n}/\log k)$, and they undercover asymptotically in the ``moderately
sparse'' regime where $s$ increases more slowly than $n$ with
$s\gg \sqrt{n}/\log k$. Indeed, \Cref{thm:l1_lower_bound} above, as well as the
results of \citet{cai_lasso_2017} and \citet{javanmard_debiasing_2018} show that
it is impossible to avoid explicitly specifying $s$ if one allows for the
moderately sparse regime.
On the other end of the spectrum, in the ``low dimensional'' regime where
$k\ll n$, the double lasso \ac{CI} is asymptotically equivalent to the usual
\ac{CI} based on the long regression. Thus, the double lasso \ac{CI} cannot be
used when the goal is to use a priori information on $\gamma$ to improve upon
the \ac{CI} based on the long regression \citep[as in, for
example,][]{muralidharan_factorial_2020}. In contrast, our approach optimally
incorporates the bound $C$ regardless of the asymptotic regime.
\section{Simulation results}\label{sec:simulation-results}
We now illustrate the performance of our methods when the penalty takes the form
of an $\ell_{1}$ norm on a subset of $k_{2}$ controls, as in \Cref{example:l1}.
We consider a design taken from \citet{belloni_inference_2014}, with data
generated from a random regressor model that supplements
\cref{eq:linear_regression_vector} with a propensity score regression
\begin{equation*}
w=Z\pi+\sigma_{\tilde{w}}\tilde{w},
\end{equation*}
with $\tilde{w}_{i}$ and $\varepsilon_{i}$ independent standard Gaussian, and
independent of $z_{i}$, which are distributed i.i.d.\ $\mathcal{N} (0, \Sigma)$ with
$\Sigma_{ij}=2^{-\abs{i-j}}$. Similar to \citet{wuthrich_omitted_2021}, we tweak
the \citet{belloni_inference_2014} design by considering regression coefficients
that are of similar magnitude rather than decaying. This allows us to separately
vary the degree of sparsity and the signal-to-noise ratio. Specifically, we set
\begin{equation*}
\gamma_{j}= \pi_{j} =
\begin{cases}
c_{1} & \text{if $j\leq k_{1}$,}\\
c_{2} & \text{if $k_{1}<j\leq k_{1}+s$,} \\
0 & \text{otherwise.}
\end{cases}.
\end{equation*}
We consider three methods for constructing \acp{CI} for $\beta$ with nominal
level 95\%. The first two methods implement \Cref{algorithm:baseline}, with the
penalty given by the $\ell_{1}$ norm of $\gamma_{2}$, the last $k_{2}$
regression coefficients. The first method, which we refer to as ``oracle,'' sets
the penalty parameter $C$ to the actual value of $\norm{\gamma_{2}}_{1}$, and
uses knowledge of the variance of the error term $\varepsilon_{i}$. The second
\ac{CI}, termed ``AKK,'' uses initial residual estimates based on the lasso
estimator (that only penalizes $\gamma_{2}$), with the penalty chosen via
10-fold cross-validation. The \ac{CI} uses the rule of thumb calibration
$C^{rot}=\norm{\widehat{\gamma}_{short}}_{1}$ from \Cref{choice_of_C_remark},
where $\widehat{\gamma}_{short}$ are \ac{OLS} estimates from a short regression
that only includes the first $k_{1}$ controls (the ``baseline'' controls). The
final method, termed ``BCH'', implements the double lasso procedure by
\citet{belloni_inference_2014}, using the R package \texttt{hdm}, without
penalizing the $k_{1}$ baseline controls and including an intercept.
The \ac{DGP} in our random regressor model depends on 8 parameters: $n$,
$k_{1}$, $k_{2}$, $s$, $\beta$, $\sigma_{\tilde{w}}$, $c_{1}$ and $c_{2}$. We
consider $n \in \{500, 1000\}$, $k_{1} \in \{5, 10\}$,
$k_{2} \in \{100, 200, 500, 1000\}$, $s\in \{10, 20, 100\}$, $\beta \in \{0, 2\}$, and
$\sigma_{\tilde{w}} \in \{0.5, 1\}$. We calibrate $c_{1}$ and $c_{2}$ by fixing
the population $R^{2}$ from the regression of $Y$ on $Z$, and fixing the ratio
$\nu_{rot} = \norm{\gamma_{2}}_{1} / \norm{\tilde{\gamma}_{short}}_{1}$. This
allows us to directly control the signal-to-noise ratio, and the validity of the
rule-of-thumb calibration: if $\nu_{rot}\leq 1$, then the population restriction
underlying our rule of thumb is valid. We consider 4 values for the population
$R^{2}$, $\{0.01, 0.1, 0.25, 0.5\}$, and 12 values for $\nu_{rot}$,
$\{0.2, 0.4, \dotsc, 2.4 \}$. This gives a total of 9,216 \acp{DGP}.
\begin{table}
\begin{threeparttable}
\caption{Simulation results for $n=500$.}\label{tab:sim_n=500}
\begin{tabular}{@{}l rrrr rrrr rrrr r@{}}
& \multicolumn{3}{c}{$k_{2} = 100$} & \multicolumn{3}{c}{$k_{2} = 200$}
& \multicolumn{3}{c}{$k_{2} = 500$} & \multicolumn{3}{c}{$k_{2} = 1000$}\\
\cmidrule(lr){2-4}\cmidrule(lr){5-7} \cmidrule(lr){8-10} \cmidrule(lr){11-13}
$\nu_{rot}$ & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} & \multicolumn{1}{@{}c@{}}{Or} & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} & \multicolumn{1}{@{}c@{}}{Or} & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} & \multicolumn{1}{@{}c@{}}{Or} & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} &
\multicolumn{1}{@{}c@{}}{Or} \\
\midrule
\multicolumn{13}{@{}c}{{Panel A\@: Coverage Probability, minimum
across DGPs}}\\
& \multicolumn{12}{c}{$s = 10$}\\
$[0,1]$ & 92.6 & 91.3 & 93.1 & 92.4 & 88.8 & 93.2 & 93.4 & 88.4 & 93.6 & 93.3 & 85.4 & 93.8 \\
$\hol{1, 1.5}$ & 91.3 & 89.2 & 93.7 & 91.1 & 86.6 & 93.8 & 89.2 & 86.9 & 93.7 & 87.4 & 84.9 & 93.9 \\
$\hol{1.5, 2.5}$ & 85.8 & 90.2 & 93.1 & 83.3 & 88.9 & 93.3 & 78.2 & 86.2 & 93.5 & 68.0 & 83.0 & 93.7 \\ [1em]
& \multicolumn{12}{c}{$s = 20$}\\
$[0,1]$ & 92.1 & 80.1 & 93.2 & 92.9 & 75.5 & 93.7 & 93.3 & 68.1 & 93.5 & 93.7 & 62.2 & 93.9 \\
$\hol{1, 1.5}$ & 92.2 & 80.9 & 94.1 & 90.7 & 74.0 & 94.1 & 90.1 & 67.9 & 93.4 & 86.6 & 58.7 & 94.6 \\
$\hol{1.5, 2.5}$ & 85.6 & 79.2 & 93.2 & 82.9 & 72.4 & 92.9 & 74.4 & 68.3 & 93.8 & 65.5 & 59.1 & 93.4 \\ [1em]
& \multicolumn{12}{c}{$s = 100$}\\
$[0,1]$ & 92.9 & 40.6 & 93.8 & 93.1 & 35.7 & 93.3 & 94.5 & 33.7 & 93.6 & 94.5 & 32.5 & 93.4 \\
$\hol{1, 1.5}$ & 92.8 & 17.1 & 93.9 & 93.9 & 14.2 & 93.9 & 93.6 & 9.8 & 93.8 & 92.6 & 8.1 & 94.7 \\
$\hol{1.5, 2.5}$ & 90.8 & 2.5 & 93.9 & 88.6 & 1.6 & 94.3 & 80.5 & 0.7 & 95.0 & 70.7 & 0.3 & 94.3 \\ [1em]
\multicolumn{13}{@{}c}{{Panel B\@: Relative length, average
across DGPs}} \\
& \multicolumn{12}{c}{$s = 10$}\\
$[0,1]$ & 1.01 & 0.96 & & 1.04 & 0.94 & & 1.10 & 0.91 & & 1.16 & 0.89 & \\
$\hol{1, 1.5}$ & 0.98 & 0.95 & & 0.99 & 0.92 & & 0.99 & 0.86 & & 1.00 & 0.82 & \\
$\hol{1.5, 2.5}$ & 0.97 & 0.95 & & 0.96 & 0.91 & & 0.95 & 0.85 & & 0.94 & 0.80 & \\ [1em]
& \multicolumn{12}{c}{$s = 20$}\\
$[0,1]$ & 1.01 & 0.96 & & 1.04 & 0.94 & & 1.10 & 0.90 & & 1.16 & 0.87 & \\
$\hol{1, 1.5}$ & 0.98 & 0.94 & & 0.98 & 0.90 & & 0.98 & 0.84 & & 0.98 & 0.79 & \\
$\hol{1.5, 2.5}$ & 0.96 & 0.94 & & 0.95 & 0.90 & & 0.92 & 0.82 & & 0.91 & 0.76 & \\ [1em]
& \multicolumn{12}{c}{$s = 100$}\\
$[0,1]$ & 1.01 & 0.95 & & 1.04 & 0.92 & & 1.10 & 0.88 & & 1.16 & 0.85 & \\
$\hol{1, 1.5}$ & 0.98 & 0.92 & & 0.98 & 0.87 & & 0.97 & 0.79 & & 0.96 & 0.73 & \\
$\hol{1.5, 2.5}$ & 0.96 & 0.91 & & 0.95 & 0.85 & & 0.90 & 0.74 & & 0.86 & 0.67 & \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item\footnotesize\emph{Notes}: For each method, panel A reports the
worst-case coverage probability of nominal 95\% level \acp{CI} over 160
\acp{DGP} for $\nu_{rot} \in [0,1]$ and $\nu_{rot} \in \hol{1.5, 2.5}$, and
64 \acp{DGP} for $\nu_{rot} \in \hol{1, 1.5}$, where each \ac{DGP} averages
across $1000$ Monte Carlo draws. Panel B reports the average relative length
across the \acp{DGP}. Relative length is defined as the average length of
the AKK and BCH \acp{CI}, averaged over the Monte Carlo draws, divided by
the average length of the oracle \ac{CI}.
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{table}
\begin{threeparttable}
\caption{Simulation results for $n=1000$}\label{tab:sim_n=1000}
\begin{tabular}{@{}l rrrr rrrr rrrr r@{}}
& \multicolumn{3}{c}{$k_{2} = 100$} & \multicolumn{3}{c}{$k_{2} = 200$}
& \multicolumn{3}{c}{$k_{2} = 500$} & \multicolumn{3}{c}{$k_{2} = 1000$}\\
\cmidrule(lr){2-4}\cmidrule(lr){5-7} \cmidrule(lr){8-10} \cmidrule(lr){11-13}
$\nu_{rot}$ & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} & \multicolumn{1}{@{}c@{}}{Or} & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} & \multicolumn{1}{@{}c@{}}{Or} & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} & \multicolumn{1}{@{}c@{}}{Or} & \multicolumn{1}{@{}c@{}}{AKK} & \multicolumn{1}{@{}c@{}}{BCH} &
\multicolumn{1}{@{}c@{}}{Or} \\
\midrule
\multicolumn{13}{@{}c}{{Panel A\@: Coverage Probability, minimum
across DGPs}}\\
& \multicolumn{12}{c}{$s = 10$}\\
$[0,1]$ & 92.8 & 91.6 & 93.3 & 93.1 & 91.1 & 93.2 & 92.3 & 90.6 & 93.1 & 92.8 & 90.1 & 92.6 \\
$\hol{1, 1.5}$ & 92.9 & 92.9 & 92.9 & 92.7 & 92.9 & 93.5 & 91.9 & 92.7 & 93.5 & 89.7 & 91.8 & 93.7 \\
$\hol{1.5, 2.5}$ & 90.0 & 91.7 & 93.0 & 88.6 & 92.2 & 92.9 & 84.0 & 91.5 & 92.6 & 79.1 & 91.4 & 93.5 \\ [1em]
& \multicolumn{12}{c}{$s = 20$}\\
$[0,1]$ & 92.8 & 86.1 & 93.0 & 93.0 & 84.3 & 92.8 & 93.6 & 79.2 & 93.4 & 93.6 & 74.9 & 93.6 \\
$\hol{1, 1.5}$ & 92.6 & 86.4 & 93.1 & 92.4 & 83.5 & 93.3 & 89.8 & 78.6 & 93.0 & 87.9 & 75.6 & 94.0 \\
$\hol{1.5, 2.5}$ & 89.4 & 86.0 & 92.9 & 89.0 & 82.4 & 93.5 & 85.0 & 78.2 & 93.2 & 76.9 & 70.1 & 92.9 \\ [1em]
& \multicolumn{12}{c}{$s = 100$}\\
$[0,1]$ & 92.9 & 27.1 & 93.2 & 93.2 & 18.6 & 93.8 & 94.1 & 13.3 & 93.7 & 94.2 & 10.3 & 93.7 \\
$\hol{1, 1.5}$ & 92.4 & 9.5 & 93.7 & 93.2 & 6.8 & 93.6 & 94.0 & 4.2 & 94.3 & 93.1 & 3.6 & 95.0 \\
$\hol{1.5, 2.5}$ & 92.0 & 2.9 & 93.0 & 91.1 & 0.8 & 92.7 & 85.8 & 0.1 & 94.2 & 75.4 & 0.0 & 94.7 \\ [1em]
\multicolumn{13}{@{}c}{{Panel B\@: Relative length, average
across \acp{DGP}}} \\
& \multicolumn{12}{c}{$s = 10$}\\
$[0,1]$ & 1.00 & 0.98 & & 1.02 & 0.96 & & 1.05 & 0.94 & & 1.09 & 0.91 & \\
$\hol{1, 1.5}$ & 0.99 & 0.97 & & 0.99 & 0.95 & & 0.98 & 0.91 & & 0.98 & 0.86 & \\
$\hol{1.5, 2.5}$ & 0.98 & 0.97 & & 0.97 & 0.95 & & 0.95 & 0.90 & & 0.93 & 0.84 & \\ [1em]
& \multicolumn{12}{c}{$s = 20$}\\
$[0,1]$ & 1.00 & 0.98 & & 1.01 & 0.96 & & 1.05 & 0.93 & & 1.09 & 0.90 & \\
$\hol{1, 1.5}$ & 0.99 & 0.97 & & 0.98 & 0.94 & & 0.98 & 0.89 & & 0.97 & 0.84 & \\
$\hol{1.5, 2.5}$ & 0.98 & 0.97 & & 0.97 & 0.94 & & 0.94 & 0.88 & & 0.91 & 0.82 & \\ [1em]
& \multicolumn{12}{c}{$s = 100$}\\
$[0,1]$ & 1.00 & 0.97 & & 1.01 & 0.95 & & 1.05 & 0.92 & & 1.09 & 0.88 & \\
$\hol{1, 1.5}$ & 0.99 & 0.95 & & 0.99 & 0.92 & & 0.98 & 0.86 & & 0.96 & 0.79 & \\
$\hol{1.5, 2.5}$ & 0.98 & 0.95 & & 0.97 & 0.91 & & 0.94 & 0.83 & & 0.89 & 0.74 & \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item\footnotesize\emph{Notes}: See \Cref{tab:sim_n=500}.
\end{tablenotes}
\end{threeparttable}
\end{table}
\Cref{tab:sim_n=500} reports the simulation results for $n=500$. The results for
$n=1000$ are reported in \Cref{tab:sim_n=1000}. In line with the theory, the
coverage of the oracle \ac{CI} is close to nominal across all
designs.\footnote{The slight undercoverage reported in the tables is due to
Monte Carlo error: with 1000 simulation draws, the expected worst-case
coverage over 160 DGPs is 93\% if the true coverage for each DGP is 95\%.}
When $\nu_{rot}\leq 1$, coverage of the AKK \ac{CI} is likewise close to
nominal. Under mild violations of the population constraint, the \acp{CI}
display moderate undercoverage: when $\nu_{rot}\leq 1.5$, coverage remains over
86.6\% across all designs, and over 90.7\% when $k_{2} \leq 200$. Only when
$\nu_{rot} > 1.5 $ and $k_{2} \geq 500$, the undercoverage becomes more severe.
In contrast, the BCH method displays moderate undercoverage even in sparse
designs with $s=10$, with coverage at about 85\% when $k_{2}=1000$ and $n=500$.
The undercoverage gets more severe, with coverage dipping below 60\% once
$s=20$, and the \acp{CI} almost entirely miss the true parameter in dense
designs with $s=100$. These results illustrate the concern discussed in
\Cref{double_lasso_comparison_sec} that asymptotic sparsity requirements may be
difficult to evaluate in finite samples.
The favorable coverage of the AKK \acp{CI} relies heavily on using the
bias-aware critical value. Unreported simulations show that the coverage of
\acp{CI} constructed using the same estimators as the AKK \acp{CI} but with
standard critical values (i.e., 1.96 for 95\% coverage), rather than our
bias-aware critical values, can be as low as 74.8\% for \acp{DGP} with
$\nu_{rot}\leq 1$.
The AKK \acp{CI} display a mild increase in average length relative to the
oracle, with the length penalty ranging between 0 and 16\%. The length penalty
relative to the BCH method is also in this range for designs where both methods
achieve good coverage. This is a bargain price to pay for the much more reliable
and transparent coverage performance.
\section{Empirical application}\label{sec:empirical-app}
This section shows the performance of our methods using survey data on $n=496$
winners of major and minor prizes in the Massachusetts lottery in 1984--88 from
\citet{imbens2001lottery} to estimate the \ac{MPE} out of unearned income, a key
structural parameter in labor and public economics. While unearned income is
typically endogenous, \citet{imbens2001lottery} argue that in this sample,
observable individual characteristics proxy well enough for the frequency of
lottery ticket purchases that the magnitude of winnings is as good as random.
The lottery winnings are paid out over 20 years, so that in a regression of the
average social security earnings in the 6 years after the lottery, $Y_{i}$, onto
yearly lottery payments, $X_{i}$, and individual controls, the coefficient on
$X_{i}$ may be interpreted as the \ac{MPE}.
We focus on a specification taken from \citet{li_linear_2020}, who augment a
baseline set of $k_{1}=7$ individual controls $Z_{1}$ consisting of the
intercept, two continuous controls (years of education and age), and 4 binary
controls (indicators for male, college, age over 55, and age over 65) with
$k_{2}=25$ additional controls $Z_{2}$ that are constructed by taking demeaned
cross-products of 4 the baseline binary controls and their interactions with
$X_{i}$ and dropping collinear terms. Both $Z_{1}$ and $Z_{2}$ are standardized.
Following the discussion in \Cref{sec:heter-treatm-effects}, the coefficient on
$X_{i}$ in this specification can be interpreted as the average \ac{MPE},
allowing for heterogeneity in the \ac{MPE} with respect to the binary controls.
In contrast, the short regression estimand in a regression that only includes
$Z_{1}$ is biased for the average \ac{MPE} in presence of such heterogeneity.
The \ac{MPE} estimate in the long regression equals $-0.049$, close to the short
regression estimate $-0.052$ that only includes the baseline controls $Z_{1}$
and corresponds to the specification in Table 4, column II row 1 in
\citet{imbens2001lottery}. However, the long regression estimate is very noisy:
the 95\% confidence interval $(-0.115, 0.016)$ includes positive values for the
average \ac{MPE} which economic theory rules out, and it is over 3 times longer
than the short regression \ac{CI} $(-0.073, -0.032)$. To increase precision of
inference, \citet{li_linear_2020} restrict the average squared mean effects
$z_{2i}'\gamma_{2}$ using an $\ell_{2}$ penalty given in \Cref{example:l2}.
Calibrating $C$ to the rule of thumb value from \Cref{choice_of_C_remark},
$C^{rot}=7.2$, yields the \ac{CI} $(-0.116, 0.018)$ using the
\citet{li_linear_2020} method, which is even longer than the long regression
\ac{CI}.\footnote{\citet{li_linear_2020} show that their method is close to
optimal in terms of weighted average length under a homoskedastic benchmark.
This may no longer be the case under heteroskedasticity. Their \ac{CI} is
variable length, and may be longer than the long regression \ac{CI} in some
samples even in the homoskedastic case. In contrast, our construction
guarantees length improvements over the long regression in all samples under
homoskedasticity.} The \ac{CI} constructed using our method,
$(-0.114, 0.015)$, improves slightly upon the long \ac{CI}, but it is still too
wide to be informative.\footnote{To make the methods more comparable and not
conflate the comparison with differences in standard error construction,
variance estimates underlying \acp{CI} for all methods use residual estimates
based on a lasso estimator that penalizes only $\gamma_{2}$, with penalty
chosen by 10-fold cross-validation.} The \citet{li_linear_2020} penalty
affords only marginal precision gains because the penalty limits the average
influence of the additional regressors---but these regressors only marginally
increase the regression $R^{2}$ in the long regression: the adjusted $R^{2}$
increases from 0.233 in the short regression to 0.236 in the long regression.
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cycle;
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\caption{95\% \acp{CI} for the marginal propensity to earn out of unearned
income under $\ell_{1}$ penalty.}\label{fig:emp}
\floatfoot{\emph{Notes:} Orange solid vertical lines at the left and right
endpoints of the $x$-axis mark \acp{CI} based on the short and long
regression, respectively. Black solid vertical line at the left endpoint of
the $x$-axis marks BCH \ac{CI}. The blue shaded area depicts the bias-aware
\acp{CI}, while the dotted black line shows the point estimates as a
function of the regularity parameter $C$. The rule of thumb value of the
regularity parameter, $C^{rot}$, its breakdown value $C^{*}$, and the
endpoint $\hat{\underline C}$ of a lower \ac{CI} for $C$, discussed in
\Cref{choice_of_C_remark}, are all marked on the $x$-axis (log scale).}
\end{figure}
In contrast, limiting the total influence of the additional controls by imposing
a bound on $\norm{\gamma_{2}}_{1}$ as in \Cref{example:l1} yields much more
substantive precision gains. \Cref{fig:emp} depicts the \acp{CI} constructed
using the implementation in \Cref{algorithm:baseline} for a wide range of the
penalty parameter. The rule of thumb calibration from \Cref{choice_of_C_remark}
yields $C^{rot}=11.8$. At this calibration the point estimate is $-0.059$, with
a \ac{CI} given by $(-0.090, -0.028)$, about half as long as the long regression
\ac{CI} (depicted by an orange vertical line in the figure). In line with the
simulation results in \Cref{sec:simulation-results}, the \ac{CI} is also close
to the double lasso \ac{CI} of \citet{belloni_inference_2014}, given by
$(-0.080, -0.031)$, (depicted in the figure by a black vertical line). Doubling
the rule-of-thumb value of $C$ changes the \ac{CI} little, yielding
$(-0.092, -0.026)$.